J Syst Sci Complex (2012) 25: 821–832

ON THE RELATIVE GENERALIZED HAMMING

WEIGHTS OF A 4-DIMENSIONAL LINEAR CODE

AND A SUBCODE WITH DIMENSION ONE∗

Zihui LIU · Wende CHEN

DOI: 10.1007/s11424-012-0192-4

Received: 5 August 2010 / Revised: 2 January 2011 c©The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2012

Abstract Finite projective geometry method is effectively used to study the relative generalized

Hamming weights of 4-dimensional linear codes, which are divided into 9 classes in order to get much more information about the relative generalized Hamming weights, and part of the relative generalized

Hamming weights of a 4-dimensional linear code with a 1-dimensional subcode are determined.

Key words Generalized Hamming weight, relative difference sequence, relative generalized Hamming weight, support weight. 1 Introduction

The relative generalized Hamming weights (RGHWs) of a linear code were first introduced in [1] by improving the wire-tap channel of type II with the coset coding scheme invented by Ozarow and Wyner[2], and they generalized the well-known generalized Hamming weights (GHWs)[3] to a two-code format. Like GHWs, RGHWs not only have usage in analyzing the coordinated multiparty wire-tap channel of type II[1], but also bring us much information about the structure of a linear code and its subcodes, so to determine GHWs and RGHWs is a basic and meaningful work.

The GHWs of linear codes with small dimensions (for example, dimension = 3, 4, 5) were extensively studied in [4–8]. In parallel with these research work, Ozarow and Wyner[2] determined all the RGHWs of a 3-dimensional linear code with its subcodes. Part of the RGHWs of a 4-dimensional linear code with its subcodes were determined in [10]. More concisely, in [10], almost all the RGHWs of a 4-dimensional linear code with a subcode with dimension greater than one were determined, and for a 4-dimensional linear code with a subcode with dimension one, only the RGHWs of the former four classes were studied according to the classification in this paper, and other classes remained unsolved. In this paper, we will study the properties of the RGHWs of other classes and determine part of the RGHWs.

Zihui LIU

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China. Email: lzhui@bit.edu.cn.

Wende CHEN

Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,

Beijing 100190, China. ∗This research is supported by the National Natural Science Foundation of China under Grant Nos. 11171366 and 61170257, and the Special Training Program of Beijing Institute of Technology. This paper was recommended for publication by Editor Xiao-Shan GAO. 822 ZIHUI LIU · WENDE CHEN 2 Finite Projective Geometry Methods 2.1 Notations and Definitions

Let J be a subset of I = {1, 2, · · · , n}. The subcode CJ of a linear [n, k] code C is defined as the set {(c1, c2, · · · , cn) ∈ C | ct = 0 for t /∈ J}. For any subcode D of C, the support of D denoted by χ(D), is defined as the set of positions where not all the codewords of D have zero coordinates. The support weight of D denoted as wS(D) is defined as follows: wS(D) = |χ(D)|.

Definition 1[3] Let C be an [n, k] linear code. The GHWs of C are defined as the parameters dr, 1 ≤ r ≤ k, where dr(C) = min{wS(D) | D is an [n, r] subcode of C}, 1 ≤ r ≤ k.

In particular, d1 is the minimum distance of C.

Definition 2[1] Let C be an [n, k] linear code and C1 be a k1-dimensional subcode of C.

Then the RGHWs of C and C1 are defined as the parameters Mj, 1 ≤ j ≤ k − k1, where

Mj = Mj(C,C1) = min{|J | | dim(CJ )− dim(C1J ) ≥ j}, 1 ≤ j ≤ k − k1.

Or equivalently,

Mj(C,C1) = min{|J | | dim(CJ )− dim(C1J ) = j}, 1 ≤ j ≤ k − k1.

Obviously, if C1 = {0}, then the GHWs of C are retrieved. RGHWs are therefore a generalization of GHWs.

In order to introduce finite geometry methods, we give another version of the definition of

RGHWs.

Definition 3[9] The RGHWs of C and C1 can also be described as the parameters Mj , 1 ≤ j ≤ k − k1, where

Mj(C,C1) = min{wS(D) | D is a subcode of C, dimD = j,D ∩ C1 = {0}} = min{wS(D) | D is a subcode of C, dimD − dimD ∩C1 = j}.

Definition 4 Assume the RGHWs of an [n, k] code C and an [n, k1] subcode C1 are (M1, · · · ,Mk−k1). Let M0 = 0, then the relative difference sequence (RDS) of C and C1 is the sequence (i0, i1,· · ·, ik−k1), where i0 = n−Mk−k1 , ij = Mk−k1−j+1 −Mk−k1−j , 1 ≤ j ≤ k − k1.

Obviously, (i0, i1,· · ·, ik−k1) and (M1, M2, · · ·, Mk−k1 , n) can be determined from each other. 2.2 Projective Subspaces and Subcodes

We describe projective geometry methods in what follows. Let u = (u1,u2,· · ·,uk) ∈ GF (q)k be a row vector (or the column vector (u1,u2,· · ·,uk)T), and L be a subset of the k coordinate positions of the vector u. Define PL(u) ∈ GF (q)k such that the t-th component of PL(u) is ut if t ∈ L and 0 if t /∈ L. An example is as follows:

Let L = {2, 4} and u = (u1, u2, u3, u4, u5) ∈GF (q)5 (or the column vector (u1,u2,u3,u4,u5)T), then PL(u) = (0, u2, 0, u4, 0) (or PL(u) = (0, u2, 0, u4, 0)T). If L = {1, 2}, then PL(u) = (u1, u2, 0, 0, 0) (or PL(u) = (u1, u2, 0, 0, 0)T).

For a subset U ⊂ GF (q)k, define PL(U) = {PL(u) | u ∈ U}. Obviously, if U is a subspace of GF (q)k, so is PL(U).

ON THE RELATIVE GENERALIZED HAMMING WEIGHTS 823

Let A be a generator matrix for a k-dimensional linear code C, then A can naturally determine a correspondence: m: GF (q)k → Z such that for any u ∈ GF (q)k, m(u) denotes the number of occurrences of u as a column in A. m(u) is called the value of u. Define the value of a subset U of GF (q)k as follows: m(U) = ∑ u∈U m(u).

Lemma 1[10] Let C be an [n, k] code, and C1 an [n, k1] subcode of C. Assume a generator matrix of C is

A = (

Ak1×n

A(k−k1)×n ) , where Ak1×n is a generator matrix for C 1. Then there is a one-one correspondence between the set of the j-dimensional subcodes D satisfying D ∩ C1 = {0} and the set of the (k − j)dimensional subspaces U ⊂ GF (q)k satisfying dim(PL(U)) = k1 such that if D corresponds to