432 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO.2. MARCH 195 Bounds on the Minimum Support Weights Tor Helleseth, Member IEEE, Torleiv Kl)ve, Senior Member, IEEE, Vladimir . Levenshtein, Member IEEE, and 0yvind Ytrehus, Member EEE - The minimum support weight, d (C). of a linear code C over GF(q) is the minimal size of the support of an r-dimensional subcode of C. number of bounds on r () are derived, generaliing the Plotkin bound and the Griesmer bound, as well as giving two new eistenial bounds. the main result, it is shown that there exist codes of any iven rate R whose ratio d/d is lower bounded by a number raning from ( q r )/(" _ q.- l ) to , depending on R. Terms-Linear codes, support weight. bounds. . INTRODUCTION F OR any code D, X(D), the support of , is the set of positions where not all the codewords of D are zero, and w s(D). the support weight of D, is the size of X()V For a linear eode C of lengh n and dimension k over GF(q') (for short: an [n, kq] code) and any integer T, where 1 � T � k, the rth minimum support weight is deined by dT = dT(C) = min{ws(D ) I D is an [n,T; q] subcode of C. In particular, the minimum distance of C is clI. The weight hierarchy of C is the set {lI, d,"',dW The rth minimum support weight is also known as the Tth generalized Ham ming weight [35]. The weight hierarchy has been called the length/dimension poile [7]. The weight hierarchy has been studied by a number of researchers in the last few years. In the Reference secion we have listed all papers, known to us, that deal with support weights and related questions. mention here one lemma that will be used later . A proof for the binary case was given in [14], the general ease ꜳ be proved similarly. Let Fk,l be the set of l-dimensional subspaces of GF(q) k. Let G be a ixed k x n generator matrix for an [n, k; q] code C. For each GF (q) k , let m(x) be the number of times appears as a column in G. For each subset of GF(q ) k, let m(U) = L m(x). E Lemma 1 For 1 � r � k we have dT(C) = n- {m(U) I U E F k , k -r} = min {m(GF(q) k \ U) I U Fk,k-r}. Manuscript received November 24, 1993; revised July 15, 1994. This research was supported by The Norwegian Research Council under Grant 463.93/005. T. HelleseLh, T. Kl0ve, and 0. Ylrehus are with the Department of Informatics. Cniversily of Bergen, HlB, N-5020 Bergen, Nor way. . L Levenshtein is with the Keldysh Instilute for Applied Mathematics, Russian Academy of Sciences. 125047 Moscow, Russia. IEEE Log Number 9408364. In this paper we give a number of bounds which generalize known bounds involving n, k, and d. A basic tool for the proof of the first two is the fundamental relation (- q)d r 2 (q" - 1)r1r-1. This was proved in [14] for q = 2, the proof given there generalizes directly to general q. Below we will give a diferent proof using a method which is of interest i n its own right. In the last and main part of the paper we give existence results. II. FUNDAMENTAL RELATION Lemma 2 For any [n, q] eode D we have L w(') = qT-I(q - l)ws(D). c'D Proof' In each position in XD), exactly q" _ "- codewords in are nonzero in that position. • For an [n, rq] code D and any 8 sueh that 0 � 8 � r, let V( D, 8 ) be the set of subspaces of D of dimension 8, the number of such subspaees i s given by the Gaussian binomial coeicients. Since wee) = ws({ae I E G(q)) the following is a generalization of Lemma 2. ea 3 For any [n, rq] code D ad any 8 such that � 8 T we have " [1'-1] ws( V ) = q r -s 8 _ 1 ws(D). VEV(D,s) Proof: By Lemma 2 we have 1 L ws (V ) = L 8-1 1 LW ( c ) VEV(D,8) VEV(D,8) q (q - ) cEV 1 q S-l (q _ 1) L _ w e e ) L 1 cED\{O} VEV(D,s) S.t. cEV 1 " [r-l] = qS_l ( q_l) � w ( e ) 8 -1 q = q S - ( � _ 1) q r -I (q - l) ws(D ) [: : � ] q = q l'-S [: : � ] ws(D). 0018-9448/95$04.00 1995 IEEE