Constructions of MDS-Convolutional Codes Ring ‘H“, with R—Nk Q@Haak

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Constructions of MDS-Convolutional Codes ring h, with rnk q@hAak. For the purpose of this correspondence, we define the rate kan convolutional code generated by q@hA Roxana Smarandache, Student Member, IEEE, as the set Heide Gluesing-Luerssen, and n k Joachim Rosenthal, Senior Member, IEEE g a fu@hAq@hA P @hAju@hA P @hAg and say that q@hA is a generator matrix for the convolutional code g q@hA qH@hA Abstract—Maximum-distance separable (MDS) convolutional codes are . If the generator matrices and both generate the same characterized through the property that the free distance attains the gener- convolutional code g then there exists a k k invertible matrix @hA alized Singleton bound. The existence of MDS convolutional codes was es- with qH@hAa @hAq@hA and we say q@hA and qH@hA are equiva- tablished by two of the authors by using methods from algebraic geometry. lent encoders. This correspondence provides an elementary construction of MDS convolutional codes for each rate and each degree . The construction is Because of this, we can assume without loss of generality that the based on a well-known connection between quasi-cyclic codes and convo- code g is presented by a minimal basic encoder q@hA. For this, let lutional codes. #i be the ith-row degree of q@hA, i.e., #i amxj deg gij @hA. In the # Index Terms—Convolutional codes, generalized Singleton bound, max- literature [7], the indexes i are also called the constraint length for the imum-distance separable (MDS) convolutional codes. ith input of the matrix q@hA. Then one defines the following. Definition 1.1: A polynomial generator matrix q@hA is called basic k # I. INTRODUCTION if it has a polynomial right inverse. It is called minimal if iaI i attains the minimal value among all generator matrices of g. The free distance of a rate kan convolutional code of degree is always upper-bounded by the generalized Singleton bound A basic generator matrix is automatically noncatastrophic, this means finite-weight codewords can only be produced from fi- dfree @n kA@ak CIAC CI (1.1) nite-weight messages. If q@hA is a minimal basic encoder one defines k the degree [8] of g as the number Xa iaI #i. In the literature, see [1]. We will provide an alternative proof of this result in the next the degree is sometimes also called the total memory [9] or the section. If aH, i.e., in the case of block codes, (1.1) simply reduces overall constraint length [7] or the complexity [10] of the minimal to the well-known Singleton bound basic generator matrix q@hA, a number dependent only on g. Among all these equivalent expressions we like the term degree best since d n k CI free (1.2) it relates naturally to equal objects appearing in systems theory and algebraic geometry. The following remarks explain this notion. cf. [2, Ch. 1, Theorem 11]. The authors of [1] showed the existence of rate kan convolutional codes of degree whose free distance was Remark 1.2: It has been shown by Forney [11] that the set equal to the generalized Singleton bound (1.1) and they called such f#IY FFFY#kg of row degrees is the same for all minimal basic codes maximum-distance separable (MDS) convolutional codes. The encoders of g. Because of this reason, McEliece [8] calls these indexes existence was established in [1] by techniques from algebraic geometry the Forney indexes of the code g. These indexes are also the same as without giving an explicit construction. This correspondence is based the Kronecker indexes of the row module on ideas from Justesen [3] and it provides an explicit construction of w a fu@hAq@hA P nhju@hA P khg MDS convolutional codes for each rate kan and each degree . The construction itself uses a well-known connection between quasi-cyclic when q@hA is a basic encoder. The Pontryagin dual of w defines a codes and convolutional codes which has been worked out by several linear time-invariant behavior in the sense of Willems [12], [13], i.e., a authors [3]–[6]. linear system. Under this identification, the Kronecker indexes of w The correspondence is structured as follows. In the remainder of this correspond to the observability indexes of the linear system [14]. The section we introduce the basic notions which will be needed throughout sum of the observability indexes is equal to the McMillan degree of the correspondence. In Section II, we give a new derivation of the gen- the system. Finally, w defines in a natural way a quotient sheaf [15] eralized Singleton bound (1.1). The main new results will be given in over the projective line and, in this context, one refers to the indexes Section III. f#IY FFFY#kg as the Grothendieck indexes of the quotient sheaf and Let be a finite field, h the polynomial ring, and @hA the field k a iaI #i as the degree of the quotient sheaf. of rational functions. Let q@hA be a k n matrix over the polynomial We feel that the degree is the single most important code parameter on the side of the transmission rate kan. In the sequel, we will adopt Manuscript received September 15, 1999; revised November 30, 2000. This the notation used by McEliece [8, p. 1082] and denote by @nY kY A a work was supported in part by NSF under Grants DMS-96-10389 and DMS-00- rate kan convolutional code of degree . 72383. The work of R. Smarandache was supported by a fellowship from the n Center of Applied Mathematics at the University of Notre Dame. J. Rosenthal For any n-component vector v P , we define its weight wt @vA as was carrying out this work while he was a Guest Professor at EPFL in Switzer- the number of all its nonzero components. The weight wt @v@hAA of a land. The material in this correspondence was presented in part at the 2000 IEEE vector v@hA P n@hA is then the sum of the weights of all its n-co- International Symposium on Information Theory, Sorrento, Italy, June 25–30, efficients. Finally, we define the free distance of the convolutional code 2000. n R. Smarandache and J. Rosenthal are with the Department of Mathematics, g& @hA through University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: Smaran- [email protected]; [email protected]). dfree aminfwt @v@hAAjv@hA PgYv@hA TaHgX (1.3) H. Gluesing-Luerssen is with the Department of Mathematics, Universität Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany (e-mail: gluesing@ It is an easy but crucial observation that in case we are given a basic mathematik.uni-oldenburg.de). encoder q@hA the free distance can also be obtained as Communicated by R. M. Roth, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(01)04751-4. dfree a minfwt @v@hAAjv@hA PwYv@hA TaHgX

This follows simply from the fact that, if q@hA has a polynomial right Once the row degrees #IY FFFY#k of the minimal basic encoder inverse, a nonpolynomial message u@hA would result in a nonpolyno- q@hA are specified one has a natural upper bound on the free distance mial codeword v@hA, which, of course, has infinite weight. of a convolutional code. The following result was derived in [1]. In the sequel, we wish to link the free distance to two types of dis- Theorem 2.1: Let g be a rate kan convolutional code generated by tances known from the literature. Following the approach in [16], [7] r a minimal-basic encoding matrix q@hA. Let #IY FFFY#k be the row we shall define the column distances dj and the row distances dj .In degrees of q@hA and # a minf#IY FFFY#kg denote the value of the order to do so let us suppose smallest row degree. Finally, let be the number of indexes #i among q@hAaq C q h C q hP C C q h# H I P # the indexes #IY FFFY#k having the value #. Then the free distance must is an encoder with row degrees #I #k. Denote by (1.4) the satisfy semi-infinite sliding generator matrix, as shown at the bottom of the dfree n@# CIA CIX (2.1) page. Then the convolutional code can be defined as k g a f@uHYuIY FFFYu Y FFFA qjuj P Y for j aHY IY FFFgX The proof given in [1] was based on the polynomial generator matrix q@hA. In the sequel, we provide a proof by means of the sliding matrix j d Then the th-order column distance j is defined as the minimum of q introduced in (1.4). v Xa @v YvY FFFYvA the weights of the truncated codewords HYj H I j re- Proof: Without loss of generality, we may assume sulting from an information sequence uHYj Xa @uHYuIY FFFYuj A with # a # # X uH TaH. Precisely, if qj denotes the k@jCIAn@jCIAupper-left sub- I k matrix of the semi-infinite matrix q, then dj aminu TaH wt@uHYj Let q be the infinite sliding generator matrix associated to q@hA as qj AX The quantity d# is called the minimum distance of the code and in (1.4). We will show that the bound (2.1) is actually a bound on the dp ad Yd Y FFFYd r the tuple H I # is called the distance profile. The 0th row distance dH defined in (1.5),; in other words, we will show that d a lim d r limit I j3I j exists and we have the relation dH n@# CIA CIX From this, the claim follows using (1.6). To dr dH dI dIX prove the bound on H, we only need to look at the first block-row of the sliding matrix q denoted by d Then I is the minimal distance computed over all finite or infinite r q aqH qI q# q# CI q# X codewords of g. It is shown in [7] that dI a dfree. H j dr H The th row distance j is defined as the minimum of the weights For all j aHY FFFY#k let qj denote the n matrix formed by v Xa @v YvY FFFYv A H H of all the finite codewords HYjC# H I jC# resulting the first rows of the matrix qj . All matrices q# CIY FFFYq# are u Xa @u Yu Y FFFYuA TaH r from an information sequence HYj H I j . Thus, zero. Hence the minimum distance dH of the n@#k CIAYk block code qr k@j CIA n@j C # CIA if we denote by j the k upper-left submatrix generated by qH qI q# q# CI q# is smaller than of the semi-infinite matrix q, the jth-row distance is the minimum distance of the n@# CIAY block code generated by r r r qH Xa qH qH qH dj a min wt@uHYj qj AX (1.5) H H I # , which is upper-bounded by the Singleton u TaH r bound n@# CIA CI. Therefore, we obtain the desired bound on dH r r The limit dI a limj3I dj exists and one has (see, e.g., [7]) for every and hence on the free distance q@hA r r r r encoder the relation dfree a dI dP dI dH n@# CIA CIX r r r dH dI dI a dfree dI dI dHX (1.6) r Remark 2.2: It was pointed out to the authors by a referee that The- In terms of state-space descriptions [17], [14] dI is equal to the min- orem 2.1 can also be derived from [8, Theorem 4.4] and [8, Corol- imal weight of a nonzero trajectory which starts from and returns to lary 4.3]. the all-zero state. dI is equal to the minimal weight of a nonzero trajectory which starts from and not necessarily returns to the all-zero In the case of a block code, i.e., when # aHand a k, the upper state. Furthermore, if the generator matrix q@hA is minimal basic, then bound in (2.1) is identical to the Singleton bound (1.2). r dI a dI a dfree (see [17], [7] for details). It follows that for a basic It is easy to see that for given nY kY and , the upper bound (2.1) is encoder the minimal-weight codewords are generated by finite infor- maximized if and only if # is as big as possible while is as small as mation sequences. possible, which results in

# a ak a #I a a # `#CI a II. THE GENERALIZED SINGLETON BOUND a #k a ak CIa# CIX (2.2) It is certainly a most natural question to ask how large the distance We will call the above set of indexes the generic set of row degrees as of a rate kan code of some bounded degree can be. McEliece [8] they are sometimes referred to in the systems literature. calls codes having the largest free distance among all @nY kY A codes distance optimal. Codes of degree aHcorrespond to nY k linear Remark 2.3: McEliece [8, p. 1083] calls a code g having the generic block codes and here we know that the distance cannot be larger than set of row degrees compact. In systems theory, the set of row degrees the Singleton bound n k CI. In [1], it was shown that the free dis- #IY FFFY#k corresponds to the observability indexes of the associated tance can never be larger than the generalized Singleton bound (1.1) (Pontryagin dual) linear system. (Compare with Remark 1.2 and [14]). for an @nY kY A-code. In the sequel we will give a new derivation of It is known that the set of all linear systems having a fixed input number this bound. k, a fixed output number n k, and a fixed McMillan degree has in a

qH qI q# q# CI q# q q q q q q a H I # # CI # X (1.4) ...... IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 5, JULY 2001 2047 natural way the structure of a smooth projective variety [15]. The subset are not necessarily the generic set of indexes. It is an open question of systems having the generic set of row degrees forms a Zariski open if there always exist convolutional codes having given row degrees subset of this variety, i.e., a generic set in the sense of algebraic geom- # a #I #k and free distance equal to the right-hand side etry. This explains why systems theorists call the indexes appearing in of (2.1). (2.2) the generic set of row degrees. We conclude the section with a simple theorem that tells us how to obtain MDS convolutional codes of rate kHan from MDS codes of rate Specializing the above result to the generic set of row degrees we get H kan where k `k. the following upper bound in terms of the degree . Theorem 2.8: Let g be a convolutional code of rate kan generated Theorem 2.4: For every base field and every rate kan convolu- kn by the minimal-basic encoding matrix q@hA P h with row tional code g of degree , the free distance is bounded by indexes dfree @n kA@ak CIAC CIX (2.3) # a #I a a # `#CI #kY where `kX q@hA P h@kIAn q@hA The main result of [1] states. Let be the matrix obtained from by omit- ting any of the last k last rows of q@hA. If the free distance of g Theorem 2.5: For any positive integers k`nYand for any prime achieves the upper bound (2.1), then the same is true for the code g p there exists a rate kan convolutional code g of degree over a suffi- generated by the encoder q. In particular, if g is an MDS code, then ciently big field of characteristic p, whose free distance is equal to the so is g. upper bound (2.3). Proof: First note that noncatastrophicity as well as the full-rank q g g Based on Theorems 2.4 and 2.5 we introduce the following notions. conditions carry over to the matrix . Moreover, the codes and both have the same minimal row degree # and the same number of Definition 2.6: The upper bound (2.3) is called the generalized Sin- rows having this degree #. Therefore, the upper bound (2.1) has the gleton bound. A rate kan code of degree whose free distance achieves same value for both codes and the theorem follows from the inclusion the generalized Singleton bound is called an MDS convolutional code. gg. The proof of Theorem 2.5 given in [1] is nonconstructive and it kan makes use of algebraic geometry. For some special set of rates kan III. A CONSTRUCTION OF RATE MDS CONVOLUTIONAL CODES and degree , e.g., k aI[18] or k a I [19], constructions which In this section, we will provide a concrete construction of an lead to MDS convolutional codes can be found in the literature. We are, @nY kY A MDS convolutional code for each degree and each rate however, not aware of a construction in the general case. kan. The underlying idea here follows the lines of [3], [5] which is The algebraic conditions used in [1] to describe the set of MDS con- an instance of the relationship between quasi-cyclic block codes and volutional codes were very involved and we do not know of a simple convolutional codes. We will not go into the details of this connection, algebraic criterion in general. For small parameters kY n and it is, rather refer the reader to [3], [4], [6]. however, often easy to decide if a particular code is MDS. The fol- As defined in [3], [5], a convolutional code is said to be generated lowing example illustrates this. by a polynomial n n n nI Example 2.7: Consider the rate PaQ convolutional code over the g@hAagH@h ACgI@h Ah C C gnI@h Ah (3.1) base field Q defined through the encoding matrix if it has a polynomial encoder of the form (3.2) shown in at the bottom II I rnk q@HA a k g@HA a g @HA TaH q@hAa X of the page. It is immediate that if H . h CI h Ph CP The code

Here the row degrees are # a #I aHand #P aI, aIand the total g a f@uH@hAY FFFYukI@hAA q@hAj aI #IY#P k degree is . form a generic set of row degrees and the upper @uH@hAY FFFYukI@hAA P hg bounds in (2.1) and (2.3) are in this case both equal to Q. It follows that q@hA is an MDS convolutional code if the free dis- is isomorphic to n n n kI tance of this code is equal to Q. One verifies that the 0th column distance f@uH@h ACuI@h Ah C C ukI@h Ah A g@hAg (3.3) dH aPand the first column distance is dI aQ, the maximal possible. the isomorphism is simply multiplexing and, therefore, weight-pre- It follows from Theorem 2.1 that MDS convolutional codes neces- serving. We will not use the description (3.3) but rather the encoder sarily have the generic set of row degrees as in (2.2). It is worth men- matrix in (3.2). tioning that within the class of all rate kan codes with fixed degree , The following theorem will lead us to the construction of MDS con- the distribution (2.2) of the row degrees leads to the smallest possible volutional codes. Recall that two elements Y P are called n-equiv- memory. n n alent if a . The set of convolutional codes of rate kan and degree is subdi- vided into codes whose encoding matrices q@hA have a fixed set of Theorem 3.1 [3, Theorem 3]: Let p be a prime and r P . Let k row degrees #IY FFFY#k with a iaI #i. In Theorem 2.1, we gave g@hA P p h generate a cyclic code over p of length x relatively an upper bound for the free distance for a code whose row degrees prime to p and of distance dg . Let n be any positive divisor of x and

gH@hA gI@hA gP@hA gnI@hA hgnI@hA gH@hA gI@hA gnP@hA hg @hA hg @hA g @hA g @hA q@hAa nP nI H nQ X (3.2) ...... hgnkCI@hA hgnkCP@hA hgnI@hA gH@hA gnk@hA 2048 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 5, JULY 2001 k`n.Ifg@hA has at most n k roots in each n-equivalence class, Next we prove that g satisfies the root condition given in Theorem then the generator matrix q@hA defined in (3.2) is basic minimal and 3.1. To do so, observe that the n-equivalence class of s, where H describes a kan convolutional code of free distance dfree ! dg . s I, consists of s sC sCP sC@nkIA sC@nkA Now we are ready to construct MDS codes of any rate kan and Y Y Y FFFY Y Y FFF X any degree . The idea is as follows. We will construct a polynomial The form of g@hA in (3.6) shows that each such n-equivalence class g@hA P p h of degree x u which generates a rate xY u contains at most n k roots of g@hA if x u @n kA. This is Reed–Solomon block code whose distance is equal to the Singleton indeed guaranteed by construction of in (3.4) bound x u CI. The parameters x and u will be chosen such that njx and dg a@n kA@ak CIAC CIY which is the MDS bound x u !ak CIC a X (3.7) for the given parameters nY kY and (see (2.3)). The polynomial g@hA n k n k will satisfy the conditions of Theorem 3.1, thus we obtain the desired q@hA MDS convolutional code. Now Theorem 3.1 implies that the encoder given in (3.2) is min- nY kY To accomplish this the following technical lemma will be needed. imal-basic and generates an MDS code with the given parameters and . Lemma 3.2: Let p be a prime and kY nY fixed positive integers such that p and n are relatively prime and k`n. Then there exist Remark 3.4: The above proof is quite similar to the proof of [3, positive integers r and Theorem 4]. Actually, Justesen’s Theorem 4 can be considered a special case of the above, namely, the case when u a k. In the above !ak CICa@n kA (3.4) construction, we have more generally u ! k, see (3.7). The case u a k @n kAj solving the Diophantine equation can occur only if , which we did not require. r n a p IX (3.5) It is interesting to study the constructed convolutional code via the semi-infinite sliding generator matrix as introduced in (1.4). To do so @ an A Proof: Consider the multiplicative group which has we expand the generator polynomial g@hA in terms of its coefficients order 0@nA. Since @pY nAaIwe know that pi0@nA Imodn for all i ! I pi0@nA I n i xu . In particular, is divisible by . Choose such that g@hAaH C Ih C C xuh X (3.4) is satisfied for xY u g@hA pi0@nA I The Reed–Solomon block code generated by has a gen- Xa X erator matrix of the form n H I xu

In the sequel, assume that Y r is a solution of (3.5) satisfying the H I xu inequality (3.4). Let x a n and let u a x @nkA@akCIA. q a . . . X (3.8) ...... It is easily seen that H `u`x. Let P p be a primitive element of p and define H I xu H I xuI g@hAa@h A@h A @h A P p hX (3.6) A direct calculation now shows that the first k rows of the matrix q appear as the upper-left corner of the matrix q in (1.4), where, again, The polynomial g@hA defines a rate xY u Reed–Solomon block code the matrix q@hA is as in (3.2). Thereafter, rows jn CIY FFFYjnC k with distance of q correspond to rows jk CIY FFFY @j CIAk of q expressing the dg a x u CIa@n kA@ak CIAC CI polynomial description (3.3). If q was an infinite sliding-block matrix q@hA as desired. it would trivially follow that the convolutional code has free distance dfree ! x u CI. Theorem 3.1 of Justesen and in particular Theorem 3.3: Let pY nY k and be integers with k`nand n not the “weight retaining property” as studied by Massey, Costello, and divisible by p. Then there exists an MDS convolutional code of rate kan Justesen [5] guarantee that the distance estimate holds for the semi- and degree over some suitably big field of characteristic p. Indeed, the infinite sliding generator matrix q. generator matrix q@hA in (3.2) induced by the polynomial g@hA given in (3.6) defines an MDS convolutional code of rate kan and degree Remark 3.5: We formulated Theorem 3.3 with a prescribed characteristic p of the field over which we construct the MDS convolu- over p . Proof: First we show that the generator matrix q@hA is of degree tional code. If one is interested in the smallest possible field where this construction works, regardless of characteristic, one should, of course, . In order to do so, we calculate the degrees of the polynomials gi@hA !akCICa@nkA in the expansion (3.1) of g@hA. First note that choose to be the smallest integer such that and n CIis a prime power. In any case, it follows immediately from deg g@hAax u a n# C n (3.4) and (3.5) that the field size is the smallest possible prime power q where # a ak and a k@akCIAbH. Since g@hA defines a for which Reed–Solomon block code it follows that all its coefficient are nonzero nP and one obtains nj@q IA q ! CPX and k@n kA (3.9) deg gi@hAa#Y for i aHY FFFYn deg gi@hAa# IY for i a n CIY FFFYn IX We close this section with a few examples. This implies that the row degrees of q@hA are indeed as in (2.2) and that q@hA is minimal. Thus, the degree of the code generated by q@hA Example 3.6: Suppose we want to construct a @QY PY SA MDS con- is simply given by the sum of the row degrees, which is in fact volutional code. The MDS bound is in this case W and from (3.9) we pr PR pr I # C@k A@# CIAak@ak CIA a X need the smallest prime power bigger than , such that is divisible by Q. The smallest possible field is S and we will need a Observe also that rnk q@HA a k. rate PRY IT Reed–Solomon code for the construction. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 5, JULY 2001 2049

PV C QSh C SUhP ICTh C RPhP V C PTh C hP q@hAa X Vh C PThP C hQ PV C QSh C SUhP ICTh C RPhP

If we want however an MDS code in characteristic P, the smallest [8] R. J. McEliece, “The algebraic theory of convolutional codes,” in Hand- field is P , and we need a rate TQY SS Reed–Solomon code. Using, book of Coding Theory, V. Pless and W. Huffman, Eds. Amsterdam, e.g., MAPLE, one calculates The Netherlands: Elsevier Science, 1998, vol. 1, pp. 1065–1138. [9] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1983. U i [10] P. Piret, Convolutional Codes, an Algebraic Approach. Cambridge, g@hAa @h A MA: MIT Press, 1988. iaH [11] G. D. Forney, “Minimal bases of rational vector spaces, with applica- a hV C RPhU C SUhT C PThS C ThR C QShQ tions to multivariable linear systems,” SIAM J. Contr., vol. 13, no. 3, pp. V P PV 493–520, 1975. C h C h C [12] J. Rosenthal, J. M. Schumacher, and E. V. York, “On behaviors and a@PV C QShQ C SUhTACh@I C ThQ C RPhTA convolutional codes,” IEEE Trans. Inform. Theory, pt. 1, vol. 42, pp. P V PT Q T 1881–1891, Sept. 1996. C h @ C h C h A [13] J. C. Willems, “Paradigms and puzzles in the theory of dynamical systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 259–294, Mar. 1991. @QY PY SA [14] J. Rosenthal, “Connections between linear systems and convolutional where is a primitive of P . Hence, an encoder for a MDS codes,” in Codes, Systems and Graphical Models, B. Marcus and J. convolutional code is given by the equation at the top of the page. Rosenthal, Eds. Berlin, Germany: Springer-Verlag, 2000, vol. 123, pp. @SY PY IPA 39–66. Example 3.7: Another example that we give is a MDS [15] M. S. Ravi and J. Rosenthal, “A smooth compactification of the space of convolutional code. The MDS bound is S@T C IA PCIa QRand, as transfer functions with fixed McMillan degree,” Acta Appl. Math, vol. before, we will need the smallest prime power pr bigger than SS, such 34, pp. 329–352, 1994. r [16] R. Johannesson and K. Zigangirov, “Distances and distance bounds for that p I is divisible by S. The smallest possible field is TI and we THY PU convolutional codes—An overview,” in Topics in Coding Theory. In need a Reed–Solomon code for the construction. honor of L. H. Zetterberg (Lecture Notes in Control and Information If we want to have the construction over a field of characteristic P we Sciences). Berlin, Germany: Springer Verlag, 1989, vol. 128, pp. V will have to take aSIin (3.5) which makes x a q IaP Ia 109–136. PSS. The Reed–Solomon code that we use has parameters x a PSS [17] J. Rosenthal and E. V. York, “BCH convolutional codes,” IEEE Trans. and u a PPP. Inform. Theory, vol. 45, pp. 1833–1844, Sept. 1999. [18] J. Justesen, “An algebraic construction of rate Ia# convolutional codes,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 577–580, Jan. 1975. IV. CONCLUSION [19] G. Lauer, “Some optimal partial-unit-memory codes,” IEEE Trans. In- form. Theory, vol. IT-25, pp. 240–243, Mar. 1979. In this correspondence, we constructed MDS convolutional codes [20] R. Smarandache, “Unit memory convolutional codes with maximum for each rate kan and for each code of degree . The construction was distance,” in Codes, Systems and Graphical Models, B. Marcus and J. based on the construction of a large Reed–Solomon block code and Rosenthal, Eds. Berlin, Germany: Springer-Verlag, 2000, vol. 123, pp. 381–396. because of this the obtained convolutional code is closely related to this Reed–Solomon code. The correspondence raises several follow-up questions. Is it possible to come up with an independent construction which does not require the relative primeness of the characteristic p and the length n of the code, and/or which does not need such large field sizes? Is it possible to carry through some subfield constructions and is it possible to come up with an algebraic decoding algorithm? Finally, it would be interesting to understand MDS convolutional codes from the point of view of state dynamics. Some answers in these directions were given in [17], [20] but more research is needed.

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