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1 TECHNION - Israel Institute of Technology Computer Science Department A CONSTRUCTION OF NON-GRS MDS by R.M. Roth and A. Lempel Technical Report #509 May 1988 CODES

2 A CONSTRUcrION OF NON-GRS MDS CODES RON M. ROTH AND ABRAHAM LEMPEL Department of Computer Science Technion, Israel Institute oftechnology Haifa Israel ABSTRACT We present a construction of long MDS codes which are not of the generalized Reed- Solomon (GRS) type. The construction employs subsets S, IS I = m, of a finite field F = GF(q) with the property that no t distinct elements ofs add up to some fixed element off. Large subsets of this kind are used to construct [n=m+,k=t+l] non-grs MDS codes over F.

3 1 I. INJ'RODUCTION An ~n,le,d] linear code over F = OF(q) is called maximum distance separable (MDS) if d = n - k + 1 [4, Ch. 11]. Some upper botmds on the lengths of MDS codes derive from the fact that, for certain ranges of k, n, and q, every MQS code must be a generalized Reed-Solomon (GRS) code, i.e., generated by <Xo al <Xn-. RS = 0. V, (), ~-I k-i k-i 1 al an- where the ai e F are distinct and V is a dia~onal, nonsingular matrix [3][5]. A "set S ~ F of size m is called an (m,t,8)-set in F if there exists an element 8 e F such that no t elements of S sum to 8. In this paper we present a construction which produces an [m+,t+1] MDS code which is not GRS from any given (m,t,8)-set in F, m ~ t +. As a method for obtaining long non-grs MDS code"s, the suggested construction reduces to the combinatorial problem of finding the largest (m,t,8)-set in OF (q ) for given t and q. Lower bounds on the cardinality of such sets are derive4 ill: Section III. For related work and references see, for instance, [1]. ll. A CONSTRUCTION OF NON-GRS CODES Let n and k be two intc!gers such th~t k ~ 3 and k + 3 S; n S;.q +. Consider the [n,ie] code over F generated by the matrix o = ~-l af-i where the ai are distinct elements of F and 8 e F. First we show that 0 does not generate a

4 GRS code; then we prove that G generates an MDS code if and only if the ai form an (n-,k-l,8)-set in F. Let k-l. gi(x)=:egijx J ~ II (x.-aj),os;is;k-l, j=o OSjst-1 : j",i and let P =[gij]osijsk-i' Consider the matrix G ~P'G = ~AA], where A= [Aij]OSijSk-1 consists of the first k columns of G. By th~ definition ofp, Aij = gi(aj ) and, therefore, A is a diagonal matrix with AU = gi(d i ) ~ O. One can easily v~rify that the-last three columns ofa are given by A = a a.,.-3 - OQ a a.,.-3 - at a a.,.-3 - ak-l where a ~ nik~l (a.,.-3 - ai) ~ 0 and b ~ 8 - :Eik~l 1 b +OQ 1 b + al 1 b + ak-l ai' To prove that G does not generate a GRS code, it suffices to show that A = [aij]' and therefore A, is not a Cauchy matrix [4, p. 33]; that is, there exist no nonzero ci and dj, distinct Xi...and distinct Yj such that Cidj au =,OS; i S; k - 1, 0 S; j S;n -k-1, x +y' & J possibly with one of the columns (rows) of A having the form doo(co ci... Ct-l)' (coo(do d l... d n - k - l»[6]. Lemma 1. [5]. Given a k x r Cauchy matrix A = [aij] over F = GF (q), we can always assume aoj = d j and alj = djyj-l, 0 S;j :S; r-l.

5 3 Applying Lemma 1 to A, after a permutation of columns and transposition, we can assume d j = 1 andyj = a- 1 (a"'_3 - aj)' 0 ~j ~ k-l, so that for some C ~ 0 and x, or c d _~J_ x +Yj C a- 1 (<Xn_3 - aj) + x = b+a., O~j~k,-I, '} = b + aj' 0 ~j ~ k - 1. Regarding (1) as a quadratic equation'in aj' it has at most < k solutions. Now, for G to generate an MDS code, every set of k columns of G must be linearly independent. It suffices to check only sets of k columns containing u = ( S)' but not Uoo = ( )'. Consider a k 'x k matrix B consisting of u and any k - 1 columns of G but Uoo' Without loss of generality, we may write B = aj- 1 af-l 1 0 Let Ai be the coefficient ofxi in the polynomial det(b (x», where B(x) = 1 1 ao 0. 1 aj- 1 k-l ak- X k-l k-l ak- X Then, det(b) = A k - + S A k _ 1 By the Vandennonde fonn ofb (x), we Qave (1) k~ k~ LAix i = IT (aj-ai) IT (x-ai)' i=o OSi <jslc- i=o k~ k~ Thus, Ak- 1 ~ 0, Ak- = -Ak- 1 L'a;, and det(b) ~ 0 if and only if L ai ~ S. Therefore, G geni=o ;=0 etates an MDS code if and only if the ai fonn an (n -,k-i,s)-set.

6 4 TIl. DERIVATION OF LOWER BOUNDS Let 8 (t,q,b) denote a largest (m,t,b)':set' in GF'(q), let M (t,q,b) = 18 (t,q,b) I, and let M (t,q) ~ max5 e F M (t,q,b). Our objective is to obtain lower bounds on M (t,q) which, by the construction of Section n, provide lower bounds on the maximal length of non-grs MDS codes. Lemma. If(t,q) = 1, then M (t,q,b) = M (t,q)for all Be GF(qj. Proof. Let BI and'~ be two distinct elements of F and let 8 I be an (m,t,bi)-set in F, where (t,q ) = 1. Consider the set 8 ~ ( a + rl(~ - B I ) a e 8 I }. Clearly, 8 is an (m,t,~)-set, implying M (t,q,b I ) ~ M (t,q,~). As B I and ~ are arbitrary elements off, the value ofm (t,q,b) is independent ofa. 0 Thus, when (t,q) = 1, it suffices to examine the values of,..say, M (t,q,0) in order to obtain lower bounds on M (t,q ). Clearly, M(l,q) =q-l for every.g, since we may set 8(I,q,0) =F - {OJ and no (m,i,o) set may contain the zero element Also, M (q -1,q ) = q -1 with 8 (q -1,q,0) = F - {a} for any a e F - (OJ. For ~ t ~ q- we distinguish between even and odd q and begin with the even case. Lemma 3. For q = h, h ~, M(,q) =q. Proof. No two distinct elements off sum to zero. 0 For a set 8 ~ F, denote by CJ(8) the sum of elements of 8. Note that every (m,t,b)-set 8, t < m, is also an (m,m-t,cj(8 )~)-set. Lemma 4. For q = h and 3 ~ t :s: i-,

7 5 Proof. M(t,q) ~!l. + i!l. if t e (3,f-) if 3 < t <!l.- Let n = {CJ>j }jh;:;l'be a basis of F = OF (q), when viewed as a vector-space of dimension hover :OF(), and associate (aoal... ah-l) e OF()h with a='ll',;/ajcj>j' Assume, first, that t is' odd. In this case the 'elements of odd Hamming weight form a (i,t,0) set, since the weight of the sum of any odd number of such elements must be odd and, therefore, nonzero. The same construction yields a (!L,t,CJ>O>-set for even t. In the special case of - t e {3,f-}, we can join the zero element to form a (I+1,t,D)-set. 0 Remark. Lemma 4 can 'be shown to hold with equality. As the full proof is rather tedious, we present here only the case t = 3. Suppose there exists a (f+,3,~)-set S. Let a e S - (~) and define T ~ S - (a). Since IT I >I' there exist distinct ~, ye T such that ~+y=~+a, implying cr({a,~,y}) = ~, in contradiction to the defmition ofs. Lemma 5. For q = h andf -1 S t S q -, M (t,q) =t +. Proof. Every (t+,,0)-set S is also a (t+,t,cr(s»-set, so that M (t,q) ~ t +. On the other hand, if there were a (t+3,t,~)-set S, it would also serve as a (t+3,3,cr(s)-<;)-set, implying t S M (3,q) - 3 = f-, contrary to the stated range of t. 0 The given lower bounds on M (t,q) guarantee the existence of [n,ie] non-grs MDS codes over OF (q), q = h, for the following values ofn and k :

8 6 k 3 4 5Sk Sf-!L-l!L Sk Sq-l n q +!L+3!l. +!L+3 k +3 These are not necessarily the longest possible codes with the said properties. For example, when q = h, h ~ 7, there exists a [q+l,4,q-] non-grs code [3, 5(3)], which can be utilized to construct [k+4,k,5] non-grs codes forf S k S q-3. We turn now to finite fields of odd size. Lemma 6. Let q be a power ofan oddprime. The,i, q + 1 M(,q)=. Proof. Let {(Xj }['~l denote the elements off so that <Xo = 0 and (Xi = -aq-i' 1SiS q~ 1, and let S = {(Xi }j~q-l). Clearly, S is a,( q;1,,o)-set in F, implying M(,q) ~ q;1. To show equality, note that any set S' r;, F of size greater than q;j must contain both (Xi and (Xq-i for some i, 1SiS q~l. 0 The following is the analog oflemma 5 for odd values ofq.!l=.!. Lemma 7. Let q be a power ofan oddprime. Then,for S t S q -, M (t,q) = t + 1. Proof. Every (t+l,i,o)-set S is also a (t+l,t,o(s»-set. The proofof tightness is similar to that in Lemma 5. 0

9 7 It is easy to see that the construction of Section IT cannot be used to obtain non-grs MDS codes for k ~ q;l. There exists however an example of a [10,5,6] non-grs construction over GF (9) [3] and it would be nice to have a general construction of non-grs MDS codes also for this range ofk.!j. Lemma 8. Let q be a power ofan oddprime. Then. for 3 S t S M (t,q ) ~ t +. Proof. Any (t+,,o)-set S is also a (t+,t,cj(s»-set. 0 In specific cases, one can do much better than that. Consider first the case q = p, an odd prime. Here we have by taking the set M (3,p) ~ Lp ; 7 J ~ r, S = { ±1, ±3, ±S,..., ±(r-l) }. The bound M (t,p) ~Lp ~ 1 + t; 1 J is easy to obtain also for larger t. For small p we have the p-3 following values ofm (t,p ), 3 S t S --: t M(t,11) M(t,13) M(t,17) M(t,19) M(t,3) Considering extension fields GF (q), q = P 11, and t S!l., we have M (t,q ) ~!l. by taking p P the elements of GF (q) whose leading coefficient is 1 when viewed as h -vectors over GF (p ). Furthermore, for 3 S t S P - 1 we have M (t,q) ~ Ll!..J.!l. by taking all h -vectors with leading t p

10 8 coefficients a, 1 S; a S;Ll!..J. t These lower bounds on M (t,q) yield non-grs MDS constructions for the following values of n and k over GF(q), q odd: for k = 3 we obtain a [q;5,3] non-grs MDS code; a special case of this construction for q == 3 (mod 4) results in a code which is known to be a complete arc: appending any column to its generator matrix violates the MDS property [, p. 15]. Applying Lemma 8, we obtain a [k+3,k,4] non-grs code for all 4 S; k S; q;1.for small values of k longer codes can be obtained; when k = 4, fo~ instance, the non-grs construction yields a code whose length is of the order t. When GF (q) is an extension field of characteristic p, [;+,k] non-grs MDS codes exist for all k S;!L - 1. p REFERENCES [1] G.T. Diderrich, H.B. Mann, "Combinatorial problems in finite Abelian groups", A Survey of Combinatorial Theory, J.N. Srivastava et al. (Ed.), North-Holland, Amsterdam, [J lw.p. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford, [3] lw.p. Hirschfeld, "Maximal sets in finite projective spaces", Surveys in Combinatorics, E.K. Lloyd (Ed.), LMS Lecture Notes Series 8, Cambridge University Press, Cambridge, 1983, pp [4] F.J. MacWilliams, NJ.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, [5] R.M. Roth, A. Lempel, "On MDS codes v~ Cauchy matrices", submitted for publication. [6] R.M. Roth, G. Seroussi, "On generator matrices ofmds codes", IEEE Transactions on Information Theory, vol. IT-31, 1985, pp