A classification of MDS binary systematic codes

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1 A classification of MDS binary systematic codes Eleonora Guerrini Department of Mathematics, University of Trento, Italy. Massimiliano Sala Boole Centre for Research in Informatics, UCC Cork, Ireland Abstract Binary MDS linear codes are well-known. We prove that any binary MDS systematic non-linear code is equivalent to a linear code and hence we classify them completely. Keywords: Non-linear, MDS codes, Hamming distance. 1 Introduction MDS codes are important because they are optimal w.r.t. their distance. However, binary MDS linear codes have been classified ([MS77]) and they are trivial. On the other hand, systematic non-linear codes can have higher distance than linear codes with the same parameters ([Pre68], [BvLW83]). The main result of this paper is to show that any binary MDS systematic non-linear code is actually equivalent to a binary MDS linear code, providing thus a complete classification. 2 Preliminaries and notation The following definitions and results are classical (see i.e. [MS77]). Definition 2.1. Let k, n N such that 1 k n, φ : (F 2 ) k (F 2 ) n be an injective function and C = Im(φ). We say that C is an (n, k) binary code. Any c C is called a word of the code. From now on code means binary code and k, n are understood. Definition 2.2. Let π be the projection π : (F 2 ) n (F 2 ) k such that π(a 1,..., a n ) = (a 1,..., a k ). We say that C is systematic if (π φ)(v) = v 26/IX/2006 BCRI preprint,

2 2 A classification of MDS binary systematic codes for any v (F 2 ) k. We say that a code C is linear, if C is a vector subspace of (F 2 ) n. We denote by C(n, k) the class of all systematic (n, k) codes, and we denote by C 0 (n, k) the subset of C(n, k) of codes containing the zero vector. Fact 2.3. Any linear code is equivalent to a code in C(n, k). Definition 2.4. For any two vectors v 1, v 2 (F 2 ) n, the Hamming distance, d(v 1, v 2 ), between v 1 and v 2 is the number of coordinates in which the two vectors differ. For any v (F 2 ) n, we denote by w(v) the weight of v, i.e the number of coordinates where v differs from zero. Let C C(n, k). The Hamming distance d of C is: d = min{d(a, b) a, b C, a b}. Let C, C C(n, k). We say that C is equivalent to C (C C ), if there exists α (F 2 ) n such that C = C + α = {c + α c C}. Fact 2.5. For C, C C(n, k), we have: If C C, then d(c) = d(c ), there is C in C 0 (n, k) such that C C. Definition 2.6. A code C in C(n, k) is MDS if d(c) = n k + 1. The following distance bound is known as the Singleton bound. Fact 2.7 ([Rom92]). Let C be a code in C(n, k). Then d(c) n k + 1. From Fact 2.7, if C is an MDS code, then C has the largest possible distance for its parameters k and n. Definition 2.8. Let C be a linear code in C(k + 1, k). We say that C is a parity-check code, if c C, where c = (c 1,..., c k+1 ), we have k+1 i=1 c i = 0. The following well-known theorem states that all the binary linear MDS codes are trivial. Theorem 2.9. There are MDS linear codes in C(n, k) only when: k = 1, i.e. C = {(0,..., 0), (1,..., 1)}, k = k + 1, i.e. C is the parity-check code, n = k, i.e. C is the whole (F 2 ) n.

3 E. Guerrini, M. Sala 3 We fix in F 2 [X, Z] = F 2 [x 1,..., x k, z 1,..., z n k ] the lexicographic order <, with x 1 < x 2... < x k < z 1 <... < z n k. If C C(n, k), then we can view C as a set of points in (F 2 ) n (F 2 ) n and we denote by I(C) its (0 dimensional) vanishing ideal in F 2 [X, Z]. We now recall the structure of the Gröbner basis of I(C). Theorem 2.10 ([GOS06], [Gue05]). Let C C(n, k) and G be the reduced Gröbner basis for I(C) w.r.t. <. Then G has the following structure: G = {x x 1,..., x 2 k + x k, z 1 f 1,..., z n k f n k }, for some f j F 2 [X], 1 j n k. Any set of polynomials endowed with the structure of Theorem 2.10 is a Gröbner basis for the ideal generated by itself, and its variety is a code in C(n, k). We can then identify such polynomial sets and codes in C(n, k), as follows. Theorem 2.11 ([GOS06],[Gue05]). Let A k,n be the set A k,n = {(f 1,..., f n k ) f j : (F 2 ) k (F 2 ), 1 j n k}. There is a bijection A k,n C(n, k), given by (f 1,..., f n k ) G = {x x 1,..., x 2 k + x k, z 1 f 1,..., z n k f n k }. 3 Characterisation of binary systematic MDS codes We begin with noting some simple facts. Let C be in C 0 (n, k), let G(C) be the reduced Gröbner basis of I(C). Then C = {(a, f 1 (a),..., f n k (a)) a (F 2 ) k }. Let F = (f 1,..., f n k ), for a code C in C(n, k) we clearly have: d(c) = n k + 1 c 1 c 2, d(c 1, c 2 ) = n k + 1 a b (Z 2 ) k, d([a, F (a)], [b, F (b)]) = n k + 1 a b (Z 2 ) k, d(a, b) + d(f (a), F (b)) = n k + 1 For a, b (Z 2 ) k such that d(a, b) = 1, we have d([f (a), F (b)]) = n k, hence f j (a) f j (b) for 1 i n k. (1) From elementary finite field theory (see i.e. [LN86]), we have that f j can be written in its Normal Form as f j = k ν=1 S {1,2,...,k} S =ν a S X S (2)

4 4 A classification of MDS binary systematic codes where X S = x i1 x i2...x is and a S = a i1 i 2...i S, if S = {i 1,..., i s }. The first result that we provide is when n = k + 1. In this case we will show that there is only one MDS code in C 0 (n, k). Lemma 3.1. If C is MDS in C 0 (k + 1, k), then C is the parity-check code. Proof. Since C is in C 0 (k + 1, k), G(C) = {x x 1,..., x 2 k + x k, z 1 + f 1 }, where f 1 is as in (2). It is easy to see that C is the parity-check code if and only if f 1 = k i=1 x i, thus we have to show that 1. a 1 = a 2 =... = a k = 1, 2. a S = 0 when S Fixed 1 j k, we consider a, b (F 2 ) k, where a = 0, and b = (b 1,..., b k ) is such that w(b) = 1, b j = 1. Since C is MDS, from (1), we have that f 1 (0) f 1 (b), i.e. f 1 (0)+f 1 (b)+1 = 0, and f 1 (b) = 1. Since w(b) = 1, f 1 (b) = a j, which implies a j = We have to prove that a S = 0 when S 1. We first show that a S = 0 in case S = 2. Let S = {s 1, s 2 }, where we have s 1, s 2 {1,..., k} and a, b (F 2 ) k such that supp(a) = S, supp(b) = {s 1 }. Since d(a, b) = 1, we have that f 1 (a) f 1 (b). As w(a) = 2 and w(b) = 1, then f 1 (b) = a s1 and f 1 (a) = a s1 + a s2 + a s1 s 2. This is equivalent to saying that a s1 + a s2 + a s1 + a s1 s 2 = 1, which means a s1 s 2 = 0. By induction on the cardinality of S, we now suppose that a S = 0 for S {1,..., k} such that S = i, and 2 i k 1. Then, we have to show that for S = S {h}, and h {1,..., k}, h S, we have a S = 0. We now consider a, b in (F 2 ) k such that supp(a) = {s 1,..., s i } = S and supp(b) = {s 1,..., s i, h} = S {h}. Since d(a, b) = 1, then f 1 (a) + f 1 (b) = 1. Since w(a) = i and w(b) = i + 1, we have f 1 (a) + f 1 (b) = ( = ( = i a T + a j ) + ( j S j S {h} a j + i+1 {h} i a T + a j ) + ( a j + a h + a S {h} + j S j S i a T + a h + a S {h} + = a h + a S {h} + i 1 l=1 i a T + a T {h}. i 1 a T ) i {h} a T {h} a T )

5 E. Guerrini, M. Sala 5 Then f 1 (a) + f 1 (b) = a S {h} + i 1 l=1 a T {h} + a h = 1. By induction hypothesis, as l 2, we have i 1 and a h = 1, so that a S {h} = 0. l=1 a T {h} = 0, The previous lemma can be restated in a more useful way. Lemma 3.2. If C C 0 (n, k) such that d(c) = 2 and C is MDS, then C is the parity-check code. We are ready for our main result. Theorem 3.3. Let C be in C 0 (n, k). If C is MDS, then C is linear. Proof. The cases k = 1 and k = n are obvious. Let now 2 k n 1. We apply (1) to a = 0 and b such that w(b) = 1. Since f j (0) = 0, then f j (b) = 1 for 1 j n k. On the other hand, since k 2, it is possible to find e (F 2 ) k such that e b and w(e) = 1. Thus d(0, e) = 1 and f j (e) = 1 for 1 j n k. Let now c 1 and c 2 be the codewords c 1 = (e, F (e)), c 2 = (b, F (b)) in (F 2 ) n. Since d(c 1, c 2 ) = 2, d(c) 2. There are two cases: either d = 2, and so C is the parity-check code by Lemma (3.1), or d = 1 and thus k = n. The following corollary is obvious. Corollary 3.4. Let C be in C(n, k). If C is MDS, then C is equivalent to a linear MDS code. 4 Conclusions We have shown that, surprisingly, the only binary MDS systematic codes are linear codes or their cosets. Binary MDS linear codes are non-interesting codes, but non-binary MDS linear codes are important in coding theory, for example the Reed-Solomon codes ([MS77]). The authors believe that there are non-binary MDS non-linear codes, which are not equivalent to a linear code and they think that this kind of techniques may be extended to attempt a classification for these codes.

6 6 A classification of MDS binary systematic codes 5 Acknowledgement Part of this work has been carried out in the Master s thesis of the first author. The first author would like to thank her supervisors: C. Traverso and the second author. For their comments and suggestions, the authors thank P. Fitzpatrick, T. Mora and J. Rosenthal. This work has been partially supported by STMicroelectronics contract Complexity issues in algebraic Coding Theory and Cryptography. References [BvLW83] Ronald D. Baker, Jacobus H. van Lint, and Richard M. Wilson, On the Preparata and Goethals codes, IEEE Trans. Inform. Theory 29 (1983), no. 3, MR MR (85c:94029) [GOS06] Eleonora Guerrini, Emmanuela Orsini, and Massimiliano Sala, Computing the distance distribution of systematic non-linear codes, BCRI preprint, 50, University College Cork, Cork, Ireland, [Gue05] [LN86] [MS77] [Pre68] [Rom92] Eleonora Guerrini, On distance and optimality in non-linear codes, Master s thesis (laurea), University of Pisa, Department of Mathematics, Rudolf Lidl and Harald Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, MR MR (88c:11073) F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, North-Holland Publishing Co., Amsterdam, 1977, North-Holland Mathematical Library, Vol. 16. MR MR (57 #5408a) Franco Preparata, A class of optimum nonlinear double-error correcting codes, Inform. Control 13 (1968), no. 13, Steven Roman, Coding and information theory, Graduate Texts in Mathematics, vol. 134, Springer-Verlag, New York, MR MR (93d:94002)