On Blocking Sets in Projective Hjelmslev Planes

Advances in Mathematics of Communications Web site: http://www.aimSciences.org Volume 1, No. 1, 2007, 65–81

ON BLOCKING SETS IN PROJECTIVE HJELMSLEV PLANES

Ivan Landjev Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev str. bl. 8 Soﬁa 1113, Bulgaria (Communicated by Thomas Honold)

Abstract. A (k,n)-blocking multiset in the projective Hjelmslev plane 3 PHG(RR) is deﬁned as a multiset K with K(P) = k, K(l) ≥ n for any line l and K(l0) = n for at least one line l0. In this paper, we investigate blocking sets in projective Hjelmslev planes over chain rings R with |R| = qm, r R/ rad R =∼ Fq, q = p , p prime. We prove that for a (k,n)-blocking multiset with 1 ≤ n ≤ q, k ≥ nqm−1(q + 1). The image of a (nqm−1(q + 1),n)-blocking multiset with n < charR under the the canonical map π(1) is “sum of lines”. In particular, the smallest (k, 1)-blocking set is the characteristic function of a line and its cardinality is k = qm−1(q + 1). We prove that if R has a subring S with p|R| elements that is a chain ring such that R is free over S then the 3 3 subplane PHG(SS ) is an irreducible 1-blocking set in PHG(RR). Corollaries 2 are derived for chain rings with |R| = q , R/ rad R =∼ Fq. 2 In case of chain rings R with |R| = q , R/ rad R =∼ Fq and n = 1, we prove that the size of the second smallest irreducible (k, 1)-blocking set is q2 + q + 1. We classify all blocking sets with this cardinality. It turns out that if charR = p there exist (up to isomorphism) two such sets; if charR = p2 the irreducible (q2 + q + 1, 1)-blocking set is unique. We introduce a class of irreducible (q2 + q + s, 1) blocking sets for every s ∈ {1,...,q + 1}. Finally, we discuss brieﬂy the codes over Fq obtained from certain blocking sets.

1. Introduction The motivation for this work comes from two sources – coding theory and finite geometry. In the past ten years, a substantial research has been done on linear codes over finite rings. It has been fuelled by the discovery that certain nonlinear codes that perform better than any linear codes over a finite field are in fact images of linear codes over the ring Z4 [8, 24]. Attempts have been made to obtain a theory of error-correcting codes over a reasonable class of rings (cf. [7, 21, 25, 28]). In [11] and [13], the theory of linear codes over the class of so-called finite chain rings was developed. Various results from [8, 11, 24, 26, 27] show that using “good” linear codes over chain rings one can construct “good” (not necessarily linear) codes over finite fields. It turns out that many good properties known for linear codes over finite fields still hold for linear codes over finite chain rings. In particular, there is an one-to-one correspondence between the classes of equivalent multisets of points in the projective

2000 Mathematics Subject Classification: Primary: 51E26, 51E21, 51E22; Secondary: 94B05. Key words and phrases: Projective Hjelmslev plane, affine Hjelmslev plane, blocking set, chain rings, Galois rings, arcs, linear codes over finite chain rings. This research has been supported by the Bulgarian NSRF under Contract MM-1405/2004.

65 c 2007 AIMS-SDU 66 I. Landjev

Hjelmslev geometries and the classes of semilinearly isomorphic fat linear codes over k the chain ring R [13]. With each multiset in PHG(RR), we can associate a linear code over R by taking a matrix with columns the points of this multiset written in homogeneous coordinates as a generator matrix. Conversely, each fat linear k code over a finite chain ring R gives a multiset of points in PHG(RR). Moreover, two multisets are equivalent if and only if the corresponding codes are semilinearly isomorphic. The task of constructing multisets of maximal cardinality in projective Hjelmslev geometries containing no more than a prescribed number of points in each hyperplane very often leads to the construction of linear codes over chain rings with interesting properties. The problem of constructing optimal multisets of points in certain finite geometries is interesting in its own right and is older than its coding theoretic counterpart. There is a vast literature about sets of points in the classical projective geometries over finite fields (cf. [9, 10] and the references there), but there are almost no results on such sets in finite Hjelmslev geometries. In this paper, we investigate 3 blocking multisets in projective Hjelmslev planes PHG(RR), where R is a chain m ring with R = q and R/ rad R = Fq. We confine ourselves to right Hjelmslev | | 3 ∼ planes PHG(RR). This is no restriction since every left R-module is a right module over the opposite ring Ropp, which is also a chain ring with qm elements and residue field of order q. In section 2, we introduce projective Hjelmslev planes and present some results on the structure of planes obtained from finite chain rings. In section 3, we define arcs and blocking multisets and present a general lower bound on the minimal size of a blocking multiset with given parameters. It is a reformulation of a upper bound m known for arcs over chain rings R with R = q , R/ rad R = Fq. In Section 4, we | | 3 ∼ m prove that for every (k,n)-blocking multiset in PHG(RR), n = 1,...,q, R = q , m 1 | | R/ rad R = F , k nq − (q + 1). In case of equality, and n < char F the image ∼ q ≥ q of the blocking set under the canonical map π(1) is a “sum of lines” (Theorem 2). Further, we prove that if R has a subring S with R elements that is a chain | | 3 ring and such that R is free over S then the subplanep PHG(SS ) is an irreducible 1- blocking set (Theorem 3). Two corollaries are derived for chain rings with R = q2, F | | R/ rad R ∼= q. In Section 5, we prove that the size of the second smallest irreducible 3 2 F 2 (k, 1)-blocking set in PHG(RR), where R = q , R/ rad R ∼= q, is q + q + 1. Moreover, in projective Hjelmslev planes| over| chain rings of characteristic p there exist (up to isomorphism) two such blocking sets, while in planes over chain rings of characteristic p2 there is only one irreducible (q2 +q+1, 1)-blocking set (Theorem 4). In section 6, we compute the parameters and the weights of various codes obtained from the linear codes associated with the blocking sets from the previous sections.

2. Projective Hjelmslev Planes We start by introducing projective Hjelmslev planes. Let Π = ( , , I) be an incidence structure. Here and are referred to as the sets of pointsP L and lines, P L respectively, and I is an incidence relation. A neighbour relation ⌣⌢ is deﬁned on and ⊆P×Lby the following conditions: P L (N1) X, Y : X ⌣⌢ Y s,t ,s = t : (X,s), (X,t), (Y,s), (Y,t) I; ∀ ∈P ⇐⇒ ∃ ∈ L 6 { }⊆ (N2) s,t : s ⌣⌢ t X, Y ,X = Y : (X,s), (X,t), (Y,s), (Y,t) I. ∀ ∈ L ⇐⇒ ∃ ∈P 6 { }⊆ A projective Hjelmslev plane is an incidence structure Π = ( , , I) with a neigh- P L bour relation ⌣⌢ that satisﬁes the axioms:

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 Blocking Sets in Hjelmslev Planes 67

(PH1) X, Y , s with (X,s) I, (Y,s) I. (PH2) ∀s,t ∈P, X∃ ∈ L, (X,s) I, (∈X,t) I; ∈ ∀ ∈ L ∃ ∈P ∈ ∈ (PH3) The relation ⌣⌢ is an equivalence relation on as well as on and P L the incidence structure ( (1), (1), J), where (1)/ ⌣⌢ is the set of classes [X] of P L P neighbour points (X ), (1)/ ⌣⌢ is the set of classes [s] of neighbour lines ∈ P L (s ), and ∈ L X ,s : ([X], [s]) J Q [X],t [s] : (Q,t) I, ∀ ∈P ∈ L ∈ ⇔ ∃ ∈ ∈ ∈ is a projective plane. An important class of projective Hjelmslev planes is obtained from the so-called finite chain rings (cf. [3, 22, 23]). Let R be finite chain ring. Denote by and P 3 , respectively, the sets of all free rank 1 and all free rank 2 submodules of RR. DefineL I by set-theoretical inclusion. It is easily checked that ( , , I) satisfies (PH1)–(PH3)⊆P×L and is a projective Hjelmslev plane. We denote this planeP L by 3 PHG(RR). Since every chain ring R is a quasi-Frobenius ring [5], the usual duality properties known from finite-dimensional vector spaces over fields hold also for R-modules. Hence every line l can be considered as the set of free rank 1 submodules 3 ∈ L (x1, x2, x3)R of RR satisfying the linear equation

r1x1 + r2x2 + r3x3 =0, where at least one of the ri’s is a unit. In other words, the dual Hjelmslev plane 3 ( , , I′) (where I′ is the transpose of I) is isomorphic to PHG(RR ). L P m Let R be a chain ring with R = q , R/ rad R = Fq. We consider the projective | | 3 ∼ Hjelmslev plane Π = ( , , I) = PHG(RR). Two points X = xR and Y = yR are called i-neighbours,Pi =L 0, 1,...,m, if X Y qi. This fact is denoted by | ∩ | ≥ X ⌣⌢ iY . Two lines s and t are i-neighbours if for every X on s there exists a point Y on t with X ⌣⌢ iY , and conversely, for every Y on t there exists X on s with Y ⌣⌢ iX. Every two points (lines) are 0-neighbours; 1-neighbourhood is the same as the neighbour relation deﬁned by (N1) and (N2). For every i 0, 1,...,m , the relation ⌣⌢ is an equivalence relation on , as ∈ { } i P well as on . The equivalence classes of this relation are denoted by [X](i), X , L ∈P respectively [s](i), s . The set of all equivalence classes of ⌣⌢ on points, resp. ∈ L i lines, is denoted by (i), resp. (i). Denote by π(i) the natural homomorphism P L π(i) : R R/Rθi, → where Rθ = rad R. Below we state some facts on the combinatorics and the structure of the projective 3 Hjelmslev planes PHG(RR) (cf. [1, 4, 6, 12, 17, 18, 19, 20, 30]). 3 m Fact 1. Let Π = ( , , I) = PHG(RR), where R is a chain ring with R = q , F P L | | and R/ rad R ∼= q. Then 2(m 1) 2 (a) = = q − (q + q + 1); |P| |L| 2(m i) (b) every point (line) has q − i-neighbours; m 1 (c) every point (line) is incident with q − (q + 1) lines (points); m i (d) given a point P and a line l there exist exactly q − points on l that are i- m i neighbours to P and exactly q − lines through P that are i-neighbours to l;

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 68 I. Landjev

(e) if X ⌣⌢ Y and X ⌣⌢ Y , X, Y , i =0, 1,...,m, there exist exactly qi lines i 6 i+1 ∈P incident with both X and Y ; by duality, if s ⌣⌢ t and s ⌣⌢ t, s,t , there i 6 i+1 ∈ L exist exactly qi points incident with both s and t; 2(i 1) 2 (f) the number of i-neighbour classes of points (lines) is q − (q + q + 1).

Deﬁne the incidence relation J (i) (i) (i) by ⊆P × L (i) (i) (i) (i) (i) ([X] , [s] ) J X′ [X] , s′ [s] : (X′,s′) I. ∈ ⇔ ∃ ∈ ∃ ∈ ∈ Fact 2. The incidence structure ( (i), (i), J (i)) is isomorphic to the projective i 3 P L (1) (1) (1) Hjelmslev plane PHG((R/Rθ ) i ). In particular, ( , , J ) is isomorphic R/Rθ P L to PG(2, q).

3 Given a projective Hjelmslev plane Π = PHG(RR), we define the affine Hjelmslev 2 plane AHG(RR) as an incidence structure with points – all points not incident with a fixed 1-neighbourhood class [l] of lines, with lines – all lines not in [l], and with incidence – the one inherited from Π. Equvalently, we define the points as all 2 pairs (a,b), a,b R, the lines as all cosets of free rank 1 submodules of RR, and ∈ 2 incidence by set-theoretical inclusion. The lines of AHG(RR) can be partitioned m 1 into q − (q + 1) classes of parallel lines. Let us fix a point P and denote by (i)(P ) the set of all lines of that ∈ P L L are incident with at least one point from [P ](i). For two lines s,t (i)(P ), we (i) ∈ L(i) write s t if they coincide on [P ] . Denote by ′ a set of lines from (P ) that contains∼ exactly one representative from each equivalenceL class under L. ∼ (i) Fact 3. The structure ([P ] , ′, I [P ](i) ′ ) is isomorphic to the affine Hjelmslev L | ×L m i 2 (m 1) − ′ plane AHG((R/Rθ − )R/Rθm−i ). In particular, ([P ] − , ′, I [P ](m 1) ) is iso- L | ×L morphic to AG(2, q).

For any X and any l , we write X ⌣⌢ l if there exists a point Y ∈ P ∈ L i ∈ P such that (Y,l) I and X ⌣⌢ iY . It is clear that in any projective Hjelmslev plane 3 ∈ Π = PHG(RR) X ⌣⌢ l π(i)(X) π(i)(l). i ⇔ ⊂ Let l be a line in Π and let i 0, 1,...,m be fixed. We define a set of points P(i) by ∈{ } (i) (m i) P = s [X] − s ,X ⌣⌢ l,s ⌣⌢ l . { ∩ | ∈ L i i } Further define an incidence relation J P(i) by ⊆ × L (m i) (m i) (s [X] − ,t) J (s [X] − ) t = . ∩ ∈ ⇔ ∩ ∩ 6 ∅ For two lines s,t , we write s t if they are incident under J with the same elements of P(i). We∈ denote L by L(i) ∼a set of lines containing exactly one representative (i) from each equivalence class of under . Set J = J P(i) L(i) . L ∼ | × Fact 4. The incidence structure (P(i), L(i), J(i)) can be embedded isomorphically m i 3 into PHG((R/Rθ − )R/Rθm−i ). Note that the structure from Fact 4 is actually isomorphic to m i 3 PHG((R/Rθ − )R/Rθm−i ) with one neighbour class of points deleted.

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 Blocking Sets in Hjelmslev Planes 69

3. Multisets of Points in Projective Hjelmslev Planes Let Π = ( , , I) be a projective Hjelmslev plane. Any mapping from the P L pointset to the nonnegative integers K : N0 is called a multiset in Π. The integerP K(P ), P , is called the multiplicityP → of P . The mapping K induces a mapping on the subsets∈P of by P K( )= K(P ), . Q Q⊆P PX ∈Q Here, the induced mapping is denoted (by a slight abuse of notation) again by K. The integer K = K( ) is called the cardinality or the size of K. A set of points with K( |)| = i isP called an i-set with respect to K. In particular, points of multiplicityQ Qi are i-points and lines of multiplicity i are i-lines. The support supp K of a multiset K is the set of points of positive multiplicity: supp K = P K(P ) > 0 . { ∈ P | } Two multisets K′ and K′′ in the projective Hjelmslev plane Π are said to be equivalent if there exists a collineation σ in Π such that K′(P ) = K′′(σ(P )) for every point P . ∈P Deﬁnition 1. A multiset K in ( , , I) is called a (k,n)-arc if P L (i) K( )= k; (ii) K(ℓP) n for every line ℓ ; ≤ ∈ L (iii) there exists at least one line ℓ0 with K(ℓ0)= n.

Deﬁnition 2. A multiset K in ( , , I) is called a (k,n)-blocking multiset if P L (i) K(P )= k; (ii) K(ℓ) n for every line ℓ ; ≥ ∈ L (iii) there exists at least one line ℓ0 with K(ℓ0)= n.

A (k,n)-blocking multiset K is called reducible if there exists (k′,n)-blocking multiset K′ with k′ < k and K′(P ) K(P ) for every point P . A blocking multiset that is not reducible is called≤minimal or irreducible. ∈ P An arc (resp. a blocking multiset) K with K(P ) 0, 1 for every P is called a projective arc (respecively, a projective blocking∈{ multiset} , or simply∈P a projective blocking set). Projective arcs and projective blocking sets can be considered as sets of points by identifying them with their support. 3 (i) A multiset K in Π = PHG(RR) induces a multiset K in the projective Hjelmslev (i) (i) i 3 plane Π = π (Π) = PHG((R/Rθ )R/Rθi ) by

(i) N , K(i) : P → 0 [P ](i) K([P ](i)). → 3 (j) The line l in PHG(RR)issaidto beoftype(i1,i2,...,im), if K (l)= ij +. . .+im, j =1 ...,m. Here ij denotes the number of points X for which there exists a point Y on l with X ⌣⌢ j Y , but for which there is no point Z on l with X ⌣⌢ j+1Z. The sequence a = (a ), (i ,i ,...,i ) Nm, where a is the number (i1,i2,...,im) 1 2 m ∈ (i1,i2,...,im) of lines of type (i1,i2,...,im), is called the spectrum of K. For any n N, we denote by κ (R3 ) the minimal value of k for which there exists ∈ n R a (k,n)-blocking multiset in PHG(R3 ). For chain rings with R = q2, R/ rad R = R | | ∼ Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 70 I. Landjev

Fq, we can reformulate the upper bound for arcs from [14, 15] to get

3 2 κn(RR) min max u(q + q + 1), ≥ 1 u q2 { ≤ ≤ q2(n 1)+ q(n u)+ u, q(q + 1)(n u/q )+ u . − − −⌊ ⌋ } If K is a projective (k,n)-arc then K′ = 1 K is a projective blocking set with parameters (q2(q2 + q + 1) k, q(q + 1) −n). Thus results about arcs can be translated into results about− blocking sets− an vice versa. Traditionally, we consider blocking sets if n is “big”, i.e. if n is “close” to the line size q(q + 1).

4. General Results on Blocking Sets in Projective Hjelmslev Planes Until the end of this section, R will be a chain ring with R = qm and R/ rad R = 2 | | ∼ Fq. As before rad R = Rθ for some θ in rad R rad R. The following theorem is based on the nested structure of projective Hjelmslev\ planes and provides a large class of blocking sets. Theorem 1. Let R be a chain ring. Let there exist blocking sets with param- i 3 m i 3 eters (k1,n1) in PHG((R/Rθ )R/Rθi ) and (k2,n2) in AHG((R/Rθ − )R/Rθm−i ), i 1,...,m 1 . Then there exists a (k k ,n n )-blocking set in P HG(R3 ). ∈{ − } 1 2 1 2 R Such blocking sets are not minimal in general. Henceforth, they are considered as trivial. It turns out that for n q we can ﬁnd the minimal size of a projective (k,n)- blocking set and provide a≤ characterization for the blocking sets with n

m Lemma 1. Let R be a chain ring with R = q , R/ rad R = Fq. Let K be a (k,n)- m 1 | | 3 ∼ blocking multiset with 1 n q − in Π = PHG(RR). Then for every neighbour ≤(1) ≤ m 1 class on points [P ] = [P ] with K([P ]) = a < q − and every neighbour class on lines [l] = [l](1) incident with [P ] in Π(1), we have m 1 K([l]) a + nq − . ≥ Proof. We use induction on m. The case m = 1 is trivial. Assume that m = 2. The structure induced on [P ] is isomorphic to AG(2, q) (cf. Fact 3). Consider the parallel class in this plane having the direction of l (i.e. lines of the form l′ [P ] ∩ where l′ [l]). Since K([P ]) < q, we have an empty line in this class. Therefore ∈ there exist q lines li [l] such that K(li [P ]) = 0. These q lines coincide on [P ] and partition the points∈ of [l] [P ]. Hence∩ \ q q K([l]) = K([P ]) + K(l [P ]) = K([P ]) + K(l ) a + nq. i \ i ≥ Xi=1 Xi=1 Now assume that our assertion has been proved for every s m 1 and every s 1 3 ≤ − s (k′,n′)-blocking multiset with n′ q − in a plane PHG(SS), where S = q , ≤ m | | S/ rad S = Fq. Further assume that R is a chain ring with q elements and residue ∼ 3 ﬁeld of order q. Set Π = PHG(RR) and let K be a (k,n)-blocking multiset in Π m 1 with n q − . Consider the incidence structure having as points all 2-neighbour classes contained≤ in [P ](1), as lines the 2-neighbour classes of lines containing points from [P ](1) and the incidence inherited from ( (2), (2), J (2)). This structure is P L Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 Blocking Sets in Hjelmslev Planes 71 isomorphic to AG(2, q) by Facts 2 and 3. The class of parallel lines in this aﬃne geometry that have the direction of l is of cardinality q and, as above, at least one m 2 of these lines has multiplicity less than q − . Let this line be incident with the points (1) [Q ](2), [Q ](2),..., [Q ](2) [P ](1). 1 2 q ⊂ (2) There exist q 2-neighbour line classes in [l], say [li] , i = 1,...,q, that meet [P ] in the points (1). These line classes partition the points of [l] [P ] and coincide on [P ]. By Fact 4, the class [l] can be embedded isomorphically\ in the projective m 1 3 (2) Hjelmslev plane Π = PHG((R/Rθ − ) m−1 ). The set [Q ] , j = 1,...,q R/Rθ { j } (2) is a 1-neighbourb class of points in Π, say [Q], and each class of lines [li] is a 1-neighbour class of lines through [Q]. By the induction hypothesis, b b (2) m 2 K([l ] b [Q]) nq − , i \ ≥ whence b q (2) m 1 K([l]) = K([P ]) + K([l ] [Q]) a + nq − . i \ ≥ Xi=1 b

3 m Lemma 2. Let Π = ( , , I) = PHG(RR), where R is a chain ring with R = q , F P L m 1 | | R/ rad R ∼= q. Let K be a (nq − (q + 1),n)-blocking set with n q. Then there exists a neighbour class on points [P ] with K([P ])=0. ≤ Proof. Assume otherwise. Let [P ] be a neighbour class on points of minimal mul- m 1 tiplicity K([P ]) = a> 0. If a q − we have ≥ m 1 2 m 1 2 m 1 K q − (q + q + 1) > q − (q + q) nq − (q + 1). | |≥ ≥ m 1 (1) If a < q − we count the multiplicities of all line classes [l] through [P ] in Π . By Lemma 1, K = K([l]) q K([P ]) − · [l]:[XP ] [l] ∈ m 1 = (q + 1)(a + nq − ) qa m 1 − m 1 = a + nq − (q + 1) > nq − (q + 1), again a contradiction.

Lemma 3. Let K be a multiset in AG(2, q) such that K(l)= n

Proof. Extend AG(2, q) to PG(2, q) and let P0 be the point on the inﬁnite line that corresponds to the missing parallel class. Deﬁne K by K(P ) if P is inb AG(2, q), K(P )=  n if P = P0,  0 otherwise. b Clearly K is a (n(q + 1),n)-blocking multiset in PG(2, q) and the result follows by Theorem 5 from [15]. b Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 72 I. Landjev

3 m Lemma 4. Let Π = ( , , I) = PHG(RR), where R is a chain ring with R = q , F P L m 1 F| | R/ rad R ∼= q. Let K be a (nq − (q + 1),n)-blocking set n

Equality is achieved for every n =1, 2,...,q. If K is a (k,n)-blocking multiset with m 1 k = nq − (q + 1), n

q (q + 1) nq − (q + 1). ≥ ≥ m 1 In order to demonstrate the existence of (nq − (q + 1),n)-blocking multisets for 3 every n = 1,...,q, fix a line l in PHG(RR). Assign multiplicity 1 to the following points: all points on l; • for every [P ](1) having a nonempty intersection with l, the points of n 1 lines • − from the affine Hjelmlsev plane induced on [P ](1) that are parallel to [P ](1) l. ∩ Clearly, every line from [l] meets l in q n points. Every line not in [l] meets some point class [X] on [l], and therefore meets≥ [l] in a single point and each one of the additional n 1 lines in the affine Hjelmlsev plane induced on [X](1) (cf. Fact 3). − 3 m 1 Let K bea(k,n)-blocking multiset in PHG(RR) with k = nq − (q +1) for some n

0 then K([P ] ) q − . Hence there exists a neighbour (1) (1) ≥ (1) m 1 class of points, [X] say, with K([X] ) = 0. By (2), K([l] ) = nq − for every 1-neighbour class of lines [l](1) through [X](1). Moreover, K(s)= n for every line s with s [X](1) = . ∩ 6 ∅ By Lemma 4, we can deﬁne a multiset H in ( (1), (1), J (1)) = PHG((R/Rθ)3 ) P L ∼ R/Rθ by

(1) N0 H : P (1) → (1) m 1 [X] K([X] )/q − . → (1) 3 For every 1-neighbour class of lines [ℓ] in PHG(RR), we have 1 H (1) K (3) ([ℓ] )= 2(m 1) (ℓ′), q − ℓ′ X[ℓ](1) ∈ m 1 (1) 2(m 1) since every point is incident with q − lines from [ℓ] . There are exactly q − lines in [ℓ](1) and thus (3) implies H([ℓ](1)) n. This gives in turn that H is an ≥ (1) (1) (1) (n(q + 1),n)-blocking set in the projective plane ( , , J ) ∼= PG(2, q). We complete the proof by applying once again TheoremP 5 fromL [15].

(1) m 1 Remark 1. It is impossible to replace the statement in this theorem that K /q − (i) m i is the sum of lines by the stronger “K /q − is the sum of lines for some i > 1”. 3 For instance, consider the following example in Π = PHG((Z8)Z8 ) which is easily generalized to the geometry over chain rings of arbitrarily high index of nilpotency. Let K be the (24, 2)-blocking set containing the following points (x1, x2, x3):

- all points from the line x1 = 0;

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 74 I. Landjev

(1) - all points on the line x2 = 0 that are not in the point class [P ] , where P = (0, 0, 1); (1) - all points on the line X1 +2x2 +2x3 = 0 that are in the point clas [P ] . 1 (2) (2) It is a straightforward check that 2 K is not the sum of two lines in Π . m 1 3 Corollary 1. Let K be a (q − (q + 1), 1)-blocking set in Π = PHG(RR), where R is a chain ring with R = qm, R/ rad R = F . Then supp K is a line in Π. | | ∼ q 1 (m 1) m 2 Proof. As in Lemma 4, the multiset K = q K − is a (q − (q + 1), 1)-blocking set (m 1) (m 1) and hence its support is a line in Π e− , [l] − say. The nonempty sets supp K (m 1) ∩ [P ] − are line segments having the same direction as l. Finally, considering the factor geometry (P(1), L(1), J(1)) deﬁned for l (Fact 4), we prove by induction on m that these line segments are collinear with the same line in Π.

Theorem 3. Let R be a chain ring with R = qm, R/ rad R = F , where qm is a | | ∼ q perfect square. Let there exist a subring S of R that is a chain ring with S = √qm and such that R is free over S. Then the multiset K deﬁned by | | 1 if P is a point from PHG(S3 ), K(P )= S 0 otherwise, 3 is a blocking set in PHG(RR). Proof. Deﬁne ξ by R = S ξS. This is possible by the fact that R is free over S. 3 ⊕ Every line in PHG(RR) can be considered as the set of points (x1, x2, x3)R that are solutions to:

(4) ax1 + bx2 + cx3 =0, where a,b,c R and at least one of them is a unit. We have to show that there exists a solution∈ to (4) with x , x , x S such that at least one of the x ’s is a 1 2 3 ∈ i unit. Without loss of generality, we set a = 1, b = s1 + ξt1, c = s2 + ξt2. By (4), we get (x1 + s1x2 + s2x3)+ ξ(t1x2 + t2x3) = 0, which is equivalent to

x1 + s1x2 + s2x3 = 0

t1x2 + t2x3 = 0 If we assume that Rt = Rθi, Rt = Rθj with 0 i j, then the submodule of R3 1 2 ≤ ≤ R generated by (1,s1,s2) and (0,t1,t2) is of shape (m,m i, 0). In other words, the m m −i 0 submodule in question is isomorphic to R/N R/N − R/N , where N = rad R and N 0 = R The dual submodule is of shape (m,i,⊕ 0) and,⊕ therefore, has a free rank 1 submodule (cf. [13]). This implies the existence of a solution with the required property. Another possibility is to set x = 1 (note that j i) and compute x and 3 ≥ 1 x2 from the linear system above. 2 F Corollary 2. Let R be a chain ring with R = q , R/ rad R ∼= q, that contains a F| | 3 subring S isomorphic to the residue field q. Then PHG(RR) contains a subplane Γ isomorphic to PG(2, q) and the projective multiset K defined by supp K =Γ is an irreducible (q2 + q +1, 1)-blocking set. The blocking sets described in Corollary 2 were introduced in the paper [2] in a slightly different context. They are defined as the orbit of a fixed point with coordinates from the residue field under a Singer cycle of PG(2, q). It turns out that the linear codes associated with these multisets can be mapped (cf. [11]) to

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 Blocking Sets in Hjelmslev Planes 75 two weight linear codes over Fq. These in turn give rise to a family of strongly regular graphs with parameters v = q6, k = q4 q, λ = q3 + q2 3q, µ = q2 q. − − − The geometric structure of these blocking sets can be easily deduced from their deﬁnition. Every point X from supp K is incident with q + 1 lines of multiplicity q +1 and on q(q + 1) (q +1) = q2 1 lines of multiplicity 1. The number of all lines blocked by points− from supp K is− (q2 + q + 1)(q + 1) + (q2 + q + 1)(q2 1) = q2(q2 + q + 1), q +1 − 3 i.e. all lines of PHG(RR) are blocked by points of supp K. This argument provides a combinatorial proof for Corollary 2.

4r Corollary 3. Let R be a chain ring with R = p , R/ rad R = F 2r , p prime. | | ∼ p Then R has a subring S with S = p2r and the projective multiset K with supp K = 3 2|r | 2r 3 PHG(SS) is an irreducible (p (p + p + 1), 1) blocking set in PHG(RR). 2r 2 F Proof. Set q = p . The rings R with R = q and R/ rad R ∼= q are either 2 | | Fq[X; σ]/(X ), σ Aut Fq, or • GR(q2,p2). ∈ • In the ﬁrst case, S is the set of all elements of the form a + bX, where a,b F . ∈ √q In the second case, S = GR(q,p2). The rest follows by Theorem 2. Remark 2. The blocking set K from Corollary 3 is not uniquely determined by its parameters. In fact, we can construct a nonisomorphic blocking set H with the same parameters as K and having K(1) = H(1). We can do this as follows. Take as supp H(1) the points of the Baer subplane of ( (1), (1), J (1)). Denote the points P L 2 (resp. the lines) of the Baer subplane by [Xi] (resp. [ℓi]) i =1,...,q + q +1. We can index the points and lines of the Baer subplane in such way that (Xi,ℓi) I. Now deﬁne the blocking set H by ∈ 1 if P [X ] ℓ for some i 1,...,q2 + q +1 , (5) H(P )= ∈ i ∩ i ∈{ } 0 otherwise. Now it is a straightforward check that H is indeed a blocking set and that every point from supp H is incident with a 1-line, i.e. the blocking set is irreducible.

5. Blocking sets with n =1. 2 F In this section, we consider chain rings R with R = q , R/ rad R ∼= q for some prime power q. It has been pointed out already that| | the classiﬁcation of such chain rings is known. If q = pr there exist exactly r +1 isomorphism classes of such rings (cf. [23],[29]). These are the Galois rings GR(q2,p2) of characteristic p2 and the 2 truncated skew polynomial rings Fq[X; σ]/(X ), σ Aut Fq (cf. [23],[29]). The 2 2 ∈ 2 commutative rings among these are GR(q ,p ) and Fq[X; id]/(X ). The blocking sets we described in the previous section have a simple structure in the sense that their image under π(1) is a trivial blocking set in PG(2, q). On the other hand, these blocking sets are not necessarily “sums of lines”. By Corollary 1, we have that the minimal size of a blocking set with n =1 is k = q(q + 1) and the support of such blocking set contains the points of a single line. A natural question is what is the size of the second smallest irreducible blocking set. Corollary 2 gives

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 76 I. Landjev a family of (q2 + q +1, 1)-blocking sets for projective Hjelmslev planes over rings that contain a subring isomorphic to their residue field. It is possible to construct another class of irreducible blocking sets with k = 2 q + q + 1, n = 1. Consider two lines ℓ and ℓ0 with ℓ ⌣⌢ ℓ0 and a point X ℓ ℓ0. Set ∈ \ 1 if P (ℓ [X]) X or P ℓ [X] (6) K(P )= ∈ 0 \ ∪{ } ∈ 1 ∩ 0 otherwise. It is readily checked that K is indeed a blocking set and that it is irreducible. This construction works for every chain ring R. It turns out that there exist no other 2 3 irreducible blocking sets of size q + q + 1 in PHG(RR). Moreover the blocking sets from (6) are the only blocking sets when the characteristic of the ring R is p2, i.e. R is the Galois ring GR(q2,p2). 2 3 Theorem 4. Let K be an irreducible (q + q + 1, 1)-blocking set in PHG(RR), 2 F R = q , R/ rad R ∼= q. Then either supp K is a projective plane of order q or else K| is| a blocking set of the type described in (6). If R = GR(q2,p2), then K is of the type described in (6). 2 3 Proof. Let K be an irreducible (q + q +1, 1)-blocking set in PHG(RR). By the irreducibility, supp K does not contain a complete line. Assume [X] is a neighbour class of points with 0 K([X]) < q. Then each class of parallel lines in the affine plane induced on [X]≤ contains an empty line (cf. Fact 3). Hence each class of neighbour lines through [X] contains at least q points outside [X]. This implies q2 + q +1= K = K([X]) + K([l] [X]) | | \ X[l] K([X]) + q(q + 1), ≥ and we have K([X]) 1. In a similar way, we can prove that K([X]) q +1. Indeed assume that K([X]) ≤ q +2. Then there is a class, [Y ] say, with K([≤Y ]) = 0. Now count the multiplicities≥ of the points in the neighbour classes of lines through [Y ]. to get the contradiction K q2 + q + 2. Thus we have proved that a neighbour |3 | ≥ class of points in PHG(RR) has multiplicity 0,1, q, or q + 1. We consider two cases: 1. There exists a neighbour class of points [X] with K([X]) = 0. Consider a neighbour class of lines, [ℓ] say, incident with [X]. In order to block all lines in [ℓ], we need at least q points. On the other hand, K([ℓ]) q + 1. Therefore, every neighbour class of lines through [X] has multiplicity q except≤ for one class of multiplicity q + 1. Furthermore, using Fact 4 together with the fact that a blocking set with q +1 or q + 2 points in a projective plane of order q does necessarily contain a line, we get that the q points in every line class through [X] are contained in the same neighbour class. For the line class with q + 1 points we can clame that q of them are neighbours while the (q + 1)-st might be in a different class. The q + 1 neighbour classes of points containing at least q points from supp K must be collinear in the factor plane. Otherwise, there is a line in the factor plane which is not blocked. Denote by [ℓ0] the neighbour class of lines incident with the q + 1 point classes of multiplicity at least q. Consider a neighbour class of points [X] with K([X]) = q. Assume there exist a point X in supp K [X] and a point X in [X] supp K such 1 ∩ 2 \ that the line X1,X2 is in [ℓ0]. There exist at least q lines through X2 (not in [ℓ0]) that have noh points fromi supp K [X]. This is a contradiction since we cannot block ∩ Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 Blocking Sets in Hjelmslev Planes 77 them by one point outside [X]. This implies that the points from [X] supp K are ∩ collinear and the q lines defined by them are in [ℓ0]. Similarly, we see that the same is true for a class with q + 1 points: q of them are incident with q neighbour lines from [ℓ0] and the (q + 1)-st point is arbitrary. Now consider incidence structure defined on [ℓ0] (cf. Fact 4). It is a isomorphic to the projective plane without a point, which we denote by P . Define a multiset H by ∞

1 if P = s [X], K(s [X]) =0, (7) H(P )= ∩ ∩ 6 0 if P = s [X], K(s [X])=0 or P = P . ∩ ∩ ∞ The multiset H is a (q +1, 1) or (q +2, 1)-blocking set in a projective plane of order q and hence contains a line. This implies that supp K either contains a complete line, in which case the blocking set is reducible, or else is of the type described in (6). 2. For every neighbour class of points K([X])=1. Consider a neighbour class of lines [ℓ] and the multiset H from (7). Since H is (q +1, 1) blocking set in a projective plane of order q the points from the classes [X] with ([X], [ℓ]) J (1) are collinear. Therefore the lines have multiplicity 1 or q + 1 and the points∈ of supp K together with the (q + 1)-lines form a projective plane of order q. Now we are going to prove that the projective Hjelmslev plane over R = GR(q2,p2) cannot contain a subplane isomorphic to a projective plane of order q. Assume otherwise and denote by ∆ a subplane isomorphic to a projective plane of order q 3 contained in Π = PHG(RR). Without loss of generality ∆ contains the points (1, 0, 0), (1, 1, 0), (1, 0, 1) and (1, 1, 1). Then it contains also (0, 1, 0), (0, 0, 1), and (0, 1, 1). Removing the neighbour class [l], where l is given by x = 0 we get − 1 an affine Hjelmslev plane Π isomorphic to AHG(R2 ). The points of ∆ l form a R \ subplane ∆ isomorphic to an affine plane of order q. Moreover, the points (0, 0), (1, 0), (0, 1) and (1, 1) are in ∆. Denote the two coordinates in Π by y and z. Then the points in ∆ on the line y = 0 are (0,ai), i =0,...,q 1, where a0 = 0, a1 =1 and a a (rad R), for i = j, in particular all a , i = 0 are− invertible. The points i 6≡ j 6 i 6 on the line y = 1 are (1,ai), i = 0,...,q 1. The intersection point of y = 0 and the line through (1,a ) parallel to (0,a− ), (1, 0) , is (0,a + a ), i.e. the set j h i i i j S = 0,a1,a2,...,aq 1 is closed under addition. Similarly, all points on the line y = z{ are (a ,a ), i −=} 0,...,q 1. The intersection point of y = 0 and the line i i − through (aj ,aj ) parallel to (0,ai), (1, 1) , i, j = 0, is (0,aiaj ). Hence S is also closed under multiplication andh S is a subringi 6 of R. But is also a subfield since S rad R = a = 0 . This is a contradiction since charR = p2. ∩ { 0} { }

3 Remark 3. The construction from (6) can be generalized. Fix a line ℓ in PHG(RR) and consider the factor geometry from Fact 4, deﬁned for the line class [ℓ]. First assume that q is even. Fix a hyperoval ℓi [Xi] i = 1,...,q +1 P , where { (1)∩ | } ∪ ∞ [Xi] are the points incident with [ℓ] in Π . We can choose ℓ to be an external line to this hyperoval and the points Xi to be incident (under I) with ℓ. For every s =0,...,q, deﬁne the multiset K by

s 1 if P (ℓ i=1[Xi]) X0,...,Xs or ∈ s\ ∪ ∪{ } (8) K(P )=  P i=0(ℓi [Xi])  0 otherwise. ∈ ∪ ∩

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2 The multiset K is an irreducible (q + q +1+ s, 1)-blocking set if every point ℓ [Xi] in the factor geometry is incident with at least two external lines to the hyperoval.∩ This is certainly true for q> 2. In the case q odd, we take ℓi [Xi] i = 1,...,q P to be an oval; ℓ is { ∩ | } ∪ ∞ again an external line to the oval and Xi are the same as for q even. The multiset K deﬁned by (8) is again an irreducible (q2 + q +1+ s, 1)-blocking set if every point ℓ [Xi] in the factor geometry is incident with at least two external lines to the oval,∩ which is true for q> 3. Note that irreducible (8, 1)- and (9, 1)-blocking sets for q = 2 and irreducible (14, 1)-, (15, 1)- and (16, 1)-blocking sets for q = 3 can easily be constructed with other methods.

6. Codes from Blocking Sets 3 It is well-known that with every multiset K in the projective geometry PHG(RR), R a chain ring, we can associate a class of isomorphic (left) linear codes over R. Every representative in this class is called a code associated with the multiset K. A code associated with K is is obtained as the module generated by the rows of a matrix with columns the points of supp K taken with the corresponding multiplicities (cf. [11]). 2 For the sake of simplicity, we consider chain rings R with R = q , R/ rad R = Fq. 3 | | ∼ Let K be a multiset in PHG(RR) with spectrum (ai0,i1 ), and let C = CK be a code associated with K. C can be mapped to a code over a q-letter alphabet (which we take as Fq) in the following way. Let T = γ0,γ1,...,γq 1 be a set of elements of R no two of which are congruent modulo{ rad R = Rθ.− Without} loss of generality, γ = 0, γ = 1. Every element r R is represented uniquely as r = r + r θ, where 0 1 ∈ 0 1 r0, r1 T . Set ∈ T F ϕ: → q , γ (γ + rad R)σ → where σ : R/ rad R F is an isomorphism. Every element r = r + r θ R can → q 0 1 ∈ be mapped to a q-tuple over Fq by 1 1 . . . 1 ψ(r) = (r1, r0) . γ0 γ1 ... γq 1 − Fk This map can be extended to q in a natural way by

ψ ((c1,c2,...,ck)) = (ψ(c1), ψ(c2),...,ψ(ck)) . Thus every (right) linear code C over the chain ring R can be mapped to a code ψ(C) over a q-ary alphabet. Generally, ψ(C) is not linear, but it can be made linear if R contains a subring isomorphic to the residue ﬁeld Fq (cf. [11]). 3 Let K be a multiset of cardinality k in PHG(RR) and let C be a code associated with K. It is easily veriﬁed that a line l produces words of the following nonzero weights in ψ(C):

if l is of type (i1,i2), i1 + i2 = k, • - q2 q words of (Hamming)6 weight qi + (q 1)(k i i ); − 1 − − 1 − 2 - q 1 words of weight q(k i1 i2); if l −is of type (i ,i ), i + i −= k,−q 1 words of weight qi . • 1 2 1 2 − 1 Below we list the parameters of some codes obtained from the blocking sets defined so far. In the case of codes over finite fields, it is typical to construct codes from the complement of a blocking set. However, in the case of codes over chain

Advances in Mathematics of Communications Volume 1, No. 1 (2007), 65–81 Blocking Sets in Hjelmslev Planes 79

parameters of the type of a line weight of the associated code codewords K (N,q ,D) i0 i1 k − i0 − i1 4 3 5 4 2 2 3 4 2 K1 (q + q ,q ,q − q ) q + q q − q 0 q − q q q2 − q q3 q4 − q2 q4 5 6 5 4 4 5 4 K2 (q ,q ,q − q ) 0 0 q q − q q5 q2 q3 − q2 q4 − q3 q5 − q4 q5 − q4 4 3 6 4 3 2 2 3 2 4 3 K3 (q + q ,q ,q − q ) q + q q − q q − q q − q q4 − q3 2q 2q2 − 2q q3 − q2 q4 − q2 q4 − q3 q 2q2 − q q3 − q2 q4 q4 − q3 q q2 − q q3 q4 − q2 q4 5 6 5 4 2 3 2 4 3 2 5 4 3 2 K4 (q ,q ,q − q − q ) 0 q − q q − q + q q − q + q − q q5 − q4 + q3 q2 − q q3 − 2q2 + q q4 − q3 + q2 q5 − q4 q5 − q4 + q3 q2 q3 − 2q2 q4 − q3 + q2 q5 − q4 − q2 q5 − q4 + q3 q2 q3 − q2 q4 − q3 q5 − q4 q5 − q4 3 2 6 3 2 2 3 2 K5 (q + q + q,q ,q − q ) q + 1 0 q q − q q3 1 q q2 q3 q3 5 4 2 2 3 4 2 5 3 K6 (q + q − q − q, q − 1 q − q q − q q − q q6,q5 − q3 − q2) q5 − q3 q2 + q − 1 q3 − 2q q4 − q2 q5 − q3 − q2 q5 − q3 3 5 2 6 1 3 1 2 3 5 2 3 K7 (q + q 2 + q ,q , q + q 2 q 2 − q 2 q q + q 2 − q − q 2 5 3 q3 + q 2 − q2 − q 2 ) q3 1 3 1 5 3 q 2 q 2 + q − q 2 q2 q3 + q 2 − q 2 q3 3 5 3 1 q − 1 q2 + q 2 q3 + q 2 − q 2 − q 5 q3 + q 2 rings, the blocking sets themselves can yield interesting codes. In all examples below, R is a chain ring with q2 elements and residue ﬁeld of order q. We set also 3 Π = PHG(RR).

2 (1) 2 1) K1: a projective (q (q + 1), q)-blocking set for which K1 is q χl, where χl is the characteristic function of a ﬁxed line from Π(1). 2) K2 =1 K1. − 2 (1) 3) K3: a projective (q (q + 1), q)-blocking set for which the support of K3 consists of q diﬀerent lines having a common point. 4) K4 =1 K3. − 2 5) K5: the (q + q +1, 1)-blocking set from Corollary 2.

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6) K =1 K . 6 − 5 7) K7: the (q(q + √q + 1), 1)-blocking set from Corollary 3. The codes obtained from K1 and K2 are MacDonald codes. For the codes obtained from K1 this was known from [11]. The codes K3 (resp. K4) though obtained from blocking sets with the same parameters as K1 (resp. K2) give rise to codes with a smaller minimum distance. The codes K5 and K6 lead to 2-weight codes and hence to strongly regular graphs [2]. The blocking set from Corollary 3 has the same spectrum as the blocking set K7 and therefore yields a code over Fq with the same parameters.

Acknowledgements The author thanks Thomas Honold for the very careful reading of the manuscript and for the numerous suggestions and remarks.

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[19] A. Kreuzer, Hjelmslev-Räume, Resultate Math., 12 (1987), 148–156. [20] A. Kreuzer, “Hjelmslevsche Inzidenzgeometrie - Ein Bericht,” Bericht TUM-M9001, Technis- che Universität München, Jan. 1990; Beiträge zur Geometrie und Algebra Nr. 17. [21] V. L. Kurakin, A. S. Kuzmin, V. Markov, A. V. Mikhalev and A. A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules (a survey), in: “Applied Algebra, Algebraic Algorithms and Error-Correcting Codes” (AAECC) 13, Lecture Notes in Computer Science, 1719 (1999), pp. 365-391. [22] B. R. McDonald, “Finite Rings with Identity,” Marcel Dekker, New York, 1974. [23] A. A. Nechaev, Finite principal ideal rings, Russian Academy of Sciences. Sbornik Mathe- matics, 20 (1973), 364–382. [24] A. A. Nechaev, Kerdock code in a cyclic form, Discrete Mathematics and Applications, 1 (1991), 365–384. [25] A. A. Nechaev, Linear codes over modules and over spaces. MacWilliams’ identity, in: “Pro- ceedings of the 1996 IEEE Int. Symp. Inf. Theory and Appl.,” Victoria B.C., Canada (1996), pp. 35-38. [26] A. A. Nechaev, Th. Honold, Weighted modules and representations of codes, Problems of Inf. Transmission, IT-35 (1999), 205–223. [27] A. A. Nechaev, A. S. Kuzmin, Linearly presentable codes, in: “Proceedings of the 1996 IEEE Int. Symp. Inf. Theory and Appl.,” Victoria B.C., Canada (1996), pp. 31-34. [28] A. A. Nechaev, A. S. Kuzmin and V. T. Markov, Linear codes over finite rings and modules, Preprint N 1995-6-1, Center of New Information Technologies, Moscow State University, 1995. [29] R. Raghavendran, Finite associative rings, Compositio Mathematica, 21 (1969), 195–229. [30] F. D. Veldkamp, Geometry over rings, in “Handbook of Incidence Geometry – Buildings and Foundations,” (ed. F. Buekenhout) chapter 19, Elsevier Science Publishers, 1995, pp. 1033-1084. Received April 2006; revised August 2006. E-mail address: [email protected]

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