Weight Hierarchies of Binary Linear Codes of Dimension 4

Abstract

AbstractThe weight hierarchy of a binary linear [n;k] code Cis the sequence (d 1 ;d 2 ;:::;d k ) whered r is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 arecollected in classes. The possible weight hierarchies in each class are given. For one class thedetails of the proofs are included. c 2001 Elsevier Science B.V. All rights reserved.Keywords: Linear code; Weight hierarchy 1. IntroductionThe weight hierarchy of linear codes has been studied by a number of researchers.For a code of dimension k, it is a sequence (d 1 ;d 2 ;:::;d k ). In particular, d 1 is theminimum distance of the code. These parameters were rst introduced in [7]. In [13]it was shown that these parameters are important in the analysis of an application oflinear codes to the wiretap channel of type II. Later, the weight hierarchy has beenshown to be important in the analysis of the trellis complexity of linear codes, seee.g. [6,9,12]; and analysis of linear codes for error detection on the local binomialchannel, see [11]. The possible weight hierarchies of binary linear codes of dimensionupto4weredeterminedin[10].In[1–3,5]westudiedthepossibleweighthierarchiesof linear codes of dimension 4 or less over arbitrary nite elds. In [1] we studiedthe weight hierarchies of codes of dimension 4; we split them into two classes: theweight hierarchies of codes satisfying the so-called chain condition and other weighthierarchies. However, this is a quite crude classication of the weight hierarchies. Inparticular, it does not tell us what are the possible weight hierarchies of codes not