# The Permutation Groups and the Equivalence of Cyclic and Quasi

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F stransitive. is ( C p uhthat such q x S = ) r eascaealinear a associate we YCLIC 1 n ecaatrz the characterize We . scci fi veriﬁes it if cyclic is x . . . , N h atsituation last The . ,j i, C H hc seither is which , } ′ . ( C n { ∈ P S ) C ODES ) n n ∈ { ′ hnits Then . 1 fpermu- of 1 ie by: given H = n , . . . , C n , . . . , n F ′ } . ( σ q . P and , ( C ) s. G ly } is ) d } p e . , simple groups. By using this classiﬁcation J. P. McSorley [24] by their composition factors [2]. Hence we have U 1 = U 2; gave the following result. this implies U1 = U2. This gives P er(C)= P ΓL(d, t). Lemma 2: A group G of degree n which is doubly tran- We recall that the group AG(n) = {τa,b : a 6= 0, (a,n) = sitive and contains a complete cycle, has as socle N, veriﬁes 1,b ∈ Z/nZ} is the group of the afﬁne transformations deﬁned N ≤ G ≤ Aut(N) and is equal to one of the cases in the as follow following Table. τa,b : Zn −→ Zn G n N (1) AGL(1,p) p Cp x 7−→ (ax + b) mod n. S4 4 C2 × C2 Sn, n ≥ 5 n Alt(n) When n = p a prime number, we have AGL(1,p)= AG(p), Alt(n), n odd and ≥ 5 n Alt(n) and is called the afﬁne group. P GL(d,t) ≤ G ≤ P ΓL(d,t) (td − 1)/t − 1 P SL(d,t) Proposition 4: Let C be a non elementary cyclic code of (d,t) =6 (2, 2), (2, 3), (2, 4) ′s length p over Fq, with q = p . Hence P er(C) is a primitive P SL(2, 11) 11 P SL(2, 11) group, and one of the following holds: M11 11 M11 M23 23 M23 1) P er(C) = PSL(2, 11) of degree 11, q = 3 ( C is the Remark 1: The projective semi-linear group P ΓL(d, t) [11, 6, 5]3 ternary Golay code or its dual), is the semi-direct product of the projective linear group 2) P er(C) = M23 of degree 23, q = 2 (C is the binary P GL(d, t) with the automorphism group Z = Gal(Ft/Fp′ ) Golay code [23, 12, 7], or its dual), ′s ′ d of ﬁnite ﬁeld Ft, where t = p ,p a prime, i.e. 3) P er(C) = P ΓL(d, t) of degree p = (t − 1)/(t − 1), where t is a power of the prime p′. P ΓL(d, t)= P GL(d, t) ⋊ Z. 4) Cp ≤ P er(C) ≤ AGL(1,p), with a normal Sylow The order of these groups are subgroup N of order p; and such that p ≥ 5. Proof: A transitive permutation group of prime degree is a |P GL(d, t)| = (d, t − 1)|PSL(d, t)|, |Z| = s primitive group [28, p. 195]. From a consequence of a result and P ΓL(d, t)= s|P GL(d, t)|. Hence if t is a prime we have of Burnside [12], we have that a transitive group of prime P ΓL(d, t)= P GL(d, t). degree is either a subgroup of the afﬁne group AGL(1,p) or The zero code, the entire space, the repetition code and its a doubly transitive group. dual are called elementary codes. The permutation group of In the doubly-transitive cases, from Lemma 2, Remark 1 these codes is Sn [21, p. 1410]. Furthermore, there is no cyclic and the Lemma 3 we have that these groups are Sp, M11, codes with permutation group equals to Alt(n). PSL(2, 11) of degree 11, M23 of degree 23, or P ΓL(d, t) of d ′s The following Lemma is a generalization of a part of [8, degree p = (t − 1)/(t − 1) over Fq, with q = p and t is a ′ Theorem E] to the non binary cases. prime power of p . From Remark 1, the group Sp corresponds Lemma 3: Let C be a non elementary cyclic code of length to an elementary code. From [25, Table 1, Lemma 2 and (J)], td−1 ′ M11 rules out, PSL(2, 11) is the permutation group of a n = t−1 over a ﬁnite ﬁeld of characteristic p , d ≥ 2, and s t = p be the power of a prime p, if the group P er(C) veriﬁes: ternary code and M23 is the permutation group of a binary code. From [21] p. 1410, these codes are unique, namely P GL(d, t) ≤ P er(C) ≤ P ΓL(d, t), PSL(2, 11) corresponds to the [11, 6, 5]3 ternary Golay code then we have, or its dual. M23 is the permutation group of the [23, 12, 7] binary Golay code and its dual. ′s d ≥ 3,t = p and P er(C)= P ΓL(d, t). Since |AGL(1,p)| = (p − 1)p, then P er(C) ≤ AGL(1,p) Proof: For d = 2. As the group P GL(2,t) acts 3- admits as order pm with m|(p − 1), hence it contains a 1 Sylow subgroup of order . From Sylow’s Theorem, transitively on 1-dimensional projective space P (Ft), we can N p N deduce from [25, Table 1 and Lemma 2], that the underlying is unique, hence it is normal. For p equals to 2 or 3, we have AGL(1,p) = S , hence the codes are elementary. Then we code is elementary, hence P er(C)= Sn, which is a contradic- p tion. Hence d ≥ 3, and similarly, we deduce from [25, Table assume p ≥ 5. ′s If (q,p)=1, the number of cyclic codes of length p over 1 and Lemma 2] that t = p . Let V denotes the permutation p−1 ord (q) +1 module of Fp′ , associated to the natural action of P GL(d, t) Fq, is equal to 2 p , where ordp(q) is the multiplicative d−1 on (d − 1) dimensional projective space P (Ft). Let U1 order of q modulo p [18]. When ordp(q)= p−1, there are only be a P GL(d, t)-submodule of V . Hence U1 is P ΓL(d, t)- four codes, namely the elementary codes. This is the cases for invariant. That is because, for σ a generator of the cyclic q = 13, p =5, 11, 31, 37, 41 or q = 11 and p = 13, 17, 23, 31. ′ σ group P ΓL(d, t)/PGL(d, t) ≃ Gal(Ft/Fp ). Then U2 = U1 , In the following table we give examples of permutation regarded as P GL(d, t)-module, is simply a twist of U1. Let groups of cyclic codes of length p over Fq in the case Fp′ be the algebraic closure of Fp′ , we conclude that the com- P er(C) ≤ AGL(1,p). The integer m is such that position factors of the Fp′ P GL(d, t)-modules U1 = Fp′ ⊗ U1 P er(C) = Cm ⋉ Cp. The codes are given in pairs, and U2 = Fp′ ⊗ U2 are the same. The submodules of the corresponding to the code and its dual. Fp′ P GL(d, t)-module V = Fp′ ⊗ V are uniquely determined q p m Codes From [8] the permutation group of the Hamming code 11 5 2 [5, 3, 3], [5, 2, 4] [15, 12, 3]2 is P ΓL(4, 2). 5 1 [5, 2, 4], [5, 3, 3] Now we will deal with the permutation group of MDS 7 3 [7, 4, 4], [7, 3, 5] codes. 19 3 [19, 16, 3], [19, 3, 6] T. P. Berger [4] proved that the only linear MDS codes with 37 6 [37, 31, 5], [37, 6, 27] minimum distance dC equals to 2 or n are the repetitions codes 13 7 2 [7, 5, 3], [7, 2, 6]; and their duals. Hence in the following result we assume that [7, 3, 5], [7, 4, 4] n > dC > 2. 17 4 [17, 13, 4], [17, 4, 12]; Theorem 8: Let C be a non elementary [n,n−dC +1, dC] ′s [17, 12, 4], [17, 5, 11]; MDS cyclic code over Fq, q = p , n > 5, n > dC > 2, [17, 8, 8], [17, 9, 7] and gcd(n − 2, dC − 2) = 1. If the group P er(C) is doubly 17 8 [17, 9, 8], [17, 8, 9] transitive, then 23 11 [23, 12, 9], [23, 11, 10] 1) P er(C)= AGL(1,p) with n = p, or ′ 29 14 [29, 15, 11], [29, 14, 12] ′ p d−1 ′ 2) P er(C)= PSL(d, p ) with n = p′−1 , d ≥ 3, gcd(p − Theorem 5: Let C be a non elementary cyclic code over 1, d)=1. F of length n = pr, where r ≥ 2 and q = p′s. Then P er(C) q Proof: If n = p, hence from Lemma 2 P er(C) is is either: i equal to AGL(1,p), PSL(2, 11), M11, M23 or a subgroup 1) an imprimitive group, and admits a system of p blocks of P ΓL(d, p′). From Proposition 4, we eliminate PSL(2, 11), each of length ps,(si = r) formed by the orbits of i M11, M23 since the [11, 6, 5]3 and [23, 12, 7]2 Golay codes are

Now, we are interested on the existence of the multiplier Proof: Let P be a Sylow p-subgroup of P er(C), hence P r Mp+1 in the group P er(C). Because it has as effect to is a p-group of Sp . From Sylow’s Theorem P is contained in a Sylow p-subgroup of Spr . It is well known that a Sylow p- simplify Algorithm 3.1 for the cyclic codes and Algorithm − − pr 1+pr 2+...+1 6.1l for extended cyclic objects [20]. subgroup of Spr is of order p . It is isomorphic Lemma 11: Let z be the largest integer such that pz|(qt − with the wreath product Zp ≀...≀Zp [28, Kaluˇznin’s Theorem], − − 1), with t order of q modulo p. Hence if z =1 we have : hence r ≤ s ≤ pr 1 + pr 2 + ... +1. r− If , from Lemma 13 we have that contains r 1 z = 1 P er(C) 1) ordp q = p t. r the group Gr, of order trp . From Lemma 11 we have that 2) The multiplier Mp+1 ∈ P er(C). r−1 2r−1 2r−1 r−1 tr = t1p , then we have |Gr| = t1p . Hence p Proof: The proof of r , follows z = 1 ⇒ ordp q = p t divides |P er(C)|, which means that P er(C) contains a p- from [18, Lemma 3.5.4]. 2r−1 r−1 subgroup of order at least p . Since z =1 ⇒ ordpr q = p t, we choose integers ak, for r kt For n

, and H(P )= AG(pr), H(P ). s−2 s−2 s−1 (b) if s > r, P = Q1 , and H(Q1 )= Q . Proposition 12: Let P be a Sylow subgroup of P er(C) − Proof: From Theorem 14 we have that r ≤ s ≤ pr 1 + which contains T . Hence the group H(P ) veriﬁes : − pr 2 + ... 1. In the case r = s, it is obvious to have P =< r 1) AG(p )= NS r (

c c The permutation τa,b can be decomposed as follows, τa,b = τT Mqj (0) = τT (0) = τ(c), τ1,bτa,0. Which gives with (7) the following equality : hence we have: l −1 −1 −1 τa,bT τa,b = τ1,bτa,0σ0 ...σl−1.τa,0 τ1,b . (8) ′ ′ ′ c ′ j τ(c)= T Mqj τ(0) = q τ(0) + c . This is well known [19, Lemma. 5.1] if σ = σ0 ...σl−1 is a ′ ′ ′ Then, j , where j . This product of the disjoint cycles and S is a permutation of Sn. τ(c)= cq + d d = c + q (τ(0) − c) − 1 − implies NS (P ) ≤ Gr. Hence SσS = S(σ0)S(σ1) ...S(σl 1). For ra = a mod l n a we obtains that τa,0(σi)= σ . This gives : Furthermore P is normal in Gr, which gives that NGr (P )= ira G . But N (P ) ≤ N (P ) ≤ N (P ) ≤ G , then r Gr P er(C) Sn r a a a la τa,0(σ0)τa,0(σ1) ...τa,0(σl−1)= σ0 σra ...σra(l−1) = T . NSn (P )= Gr. For rb = b mod l, we obtain From the Lemma 12 we have that NSn (P ) ⊂ H(P ). Hence NS (P )= Gr ⊂ H(P ). −1 n τ1,bσiτ = τi+r , Now, let τ ∈ H(P ), hence there exist c and j such that : 1,b b −1 c c hence τT τ = T Mqj ⇐⇒ τT = T Mqj τ. l−1 l −1 l That is because P ≤ Gr. For 0 we obtains : τ1,bT τ1,b = Y σi+rb = T . τT (0) = τ(1) = qj τ(0) + c. i=0 c j τT (1) = τ(2) = T Mqj (τ(1)) = q τ(1) + c c j Finally we obtains : τT (2) = τ(3) = T Mqj (τ(2)) = q τ(2) + c . l −1 l −1 −1 la −1 la . τa,bT τa,b = τ1,bτa,0T τa,0 τ1,b = τ1,bT τ1,b = T . τ(i)= qij (τ(0)) + q(i−1)j c + q(i−2)j c + ... + c. l Then the elements of H(P ) are τ ∈ Sn such that τ(i) = This gives α = a, hence τa,b ∈ NSn (

Corollary 20: The set H′(P ) veriﬁes AG(n) ⊂ H′(P ). ACKNOWLEDGMENT

Proof: From Proposition 17 we have AG(n) ≤ NSn (< The author would like to thank Professor Thierry P. Berger l T >). Furthermore, from Proposition 19 we have NSn (< for help and encouragement in carrying out the research in T l >) ⊂ H′(P ), hence the result. this paper. By using the software GAP, we ﬁnd on several examples REFERENCES that the set H′(P ) is an imprimitive group. By using this conjecture we prove the following result. [1] B . Alspach and T. D. Parson, Isomorphism of circulant graphs and Proposition 21: If n is even then, H′(P ) is either digraphs, Discrete Math. 25, 1979. Theory, 43(5), Sep. 1997. [2] M. Bardoe and Sin, The permutation modules for GL(n + 1, Fq) acting n n+1 1) imprimitive, on P (Fq ) and Fq , J. London Math. Soc. 61, 58-80, 2000. [3] T. P. Berger, A direct proof for the automorphism group of Reed–Solomon codes Proceedings of Eurocode 90 (G. Cohen and P. Charpin ed. ), Lecture notes in Computer science, Springer-Verlag 514, Berlin, Heidelberg, New York, 21-29, 1991. [4] T. P. Berger, Groupes de permutations des codes MDS afﬁne-invariants Communication in Algebra. 21(1), 239-254, 1993. [5] T. P. Berger, On the automorphism groups of afﬁne invariant codes Designs, codes and cryptography. 7, 215-221, 1996. [6] T. P. Berger and P. Charpin The permutation group of afﬁne invariant extended cyclic codes IEEE Trans. Inform. Theory, 62(6), Nov. 1996. [7] T. P. Berger and P. Charpin The automorphism groups of BCH codes and off some-afﬁne invariant codes over extension ﬁelds, Designs, Codes and Cryptography. 18, 29-53, 1999. [8] R. Bienert and B. Klopsch Automorphism group of cyclic codes J. Algebraic. Combin. 31, 33-52, 2010. [9] N. Brand, Polynomial isomorphisms of combinatorial objects, Graphs Combin. (7), 7-14, 1991. [10] J. Brillhart, J. Tonascia and P. Weinberger, On the Fermat quotient, in “ Computer in number theory” A. O. L. Atkin and B. Birch, Eds., Academic press, New york, 1991. [11] W. Burnside, On some properties of groups of odd order, J. London. Math. Soc, 33, 162-185, 1901. [12] W. Burnside, theory of groups of ﬁnite order, Dover, 1955. [13] P. J. Cameron, Finite permutation group and ﬁnite simple groups Bull. London Math. Soc. 13, 1981. [14] C. Chabot, Reconnaissance de codes, structure des codes quasi- cycliques, Ph.D. thesis, Xlim, Limoges University, Sep. 2009. [15] J. H. Conway, S. J. Lomonaco Jr and N. J. A. Sloane, A [45, 13] code with minimum distance 16 . Discrete Math. (83), 2, 213-217, 1990. [16] J. D. Dixon and B. Mortimer Permutation groups , Grad Texts Math (163), Springer-Verlag, 1996. [17] E. D. Dobson, On groups of odd prime-power degree that contain a full cycle, Discrete. Math (299), 65-78, 2005. [18] K. Guenda, Sur l’´equivalence des Codes, Ph.D. thesis submitted to the University of Science and Technology, U.S.T.H.B, Algiers, Algeria. [19] M. Hall, Jr. The theory of groups, The Macwilliams Company, New York, 1970. [20] W. C. Huffman, V. Job and V. Pless, Multiplier and generalized multipliers of cyclic objects and cyclic codes Journal of Combin. Th A (62) 183-215 1993. [21] W. C. Huffman, Codes and groups, in: V. S. Pless, W. C. Huffman (Eds.), Handbook of coding theory, Vol. II, Elsevier, Amsterdam, 1345- 1439, 1998. [22] J. S. Leon, J. M. Masley and V. Pless, Duadic codes, IEEE Trans. Inform. Theory, 30(5), Sep. 1984. [23] F. J. Macwilliams and N. J. A Sloane, The theory of error correcting- codes, Benjamin, Inc. Amsterdam, North-Holland, 1977. [24] J. P. McSorley, Cyclic permutation groups in doubly transitive groups, Comm. Algebra, 25, 33-35, 1997. [25] B. Mortimer, The modular permutation representation of the known doubly transitive groups, Proc. London Math. Soc. 41, 1-20, 1980 [26] A. Otmani, J. P. Tillich and L. Dallot, Cryptanalysis of a McEliece Cryp- tosystem based on quasi-cyclic codes, Proceeding of the First Conference on Symbolic computation and Cryptography (SCC 2008), Beijing, China, p. 69-81, April, 2008. [27] P. P. Palfy, Isomorphism problem for relational structures with a cyclic automorphism, Europ. J. Combin. 8, 35-43, 87. [28] D. J. S. Robinson, A course in the theory of groups, Grad Texts Math (80), Springer-Verlag, 1980. [29] R. Roth and G. Seroussi, On cyclic MDS codes of length q over GF(q), IEEE Trans. Inform. Theory, Vol (I), 32 March 1986. [30] H. Wienlandt, Finite permutation group, Academic Press, New York, 1964. Springer-Verlag, 1980. [31] A. Williamson, On primitive permutation groups containing a cycle, Math. Z. 130, 159-162 1973.