The system of weight equations for a binary (n, m)-code with respect to its ordered basis is introduced. It connects certain quantities (characteristics) related to the basis with the weights of non-zero words in the code. It is shown that the portion involving the variables does not depend neither on the code nor on the basis. | Turk J Math 23 (1999) , 465 – 483. ¨ ITAK ˙ c TUB THE WEIGHT EQUATIONS FOR BINARY LINEAR CODES∗ ˙ ˙ Ersan Akyıldız, Ismail S ¸ . G¨ ulo˘glu & Masatoshi Ikeda Abstract The system of weight equations for a binary (n, m)-code with respect to its ordered basis is introduced. It connects certain quantities (characteristics) related to the basis with the weights of non-zero words in the code. It is shown that the portion involving the variables does not depend neither on the code nor on the basis. Explicit forms of the matrix of coefficients in the system and its inverse matrix are computed. 1. Introduction The aim of this note is to introduce the system of weight equations for a binary (n, m)code C wiht respect to an ordered basis B of C. It consists of 2m − 1 linear equations involving 2m − 1 variables, and connects certain quantities (characteristics) related to the basis with the weights of non-zero words of C. The procedure to obtain the system is in fact elementary but rather tedious, hence it is deliberately prefered to proceed in a clumsy way (see Section 2). It turns out that the portion of the system involving the variables depends only on the dimension m of the code C, but independent from the choice of C, or of the basis B (Theorem 1). Then explicit forms of the matrix of the coefficients (in the system) and its inverse matrix are calculated (Theorems 2 and 3). As an application a necessary and sufficient condition for a finite sequence of positive integers to be a well-arranged weight pattern of an (n, m)-code is derived. ∗ This ¨ ITAK, ˙ work was carried out at the Marmara Research Center, TUB in the frame of the research project “Coding Theory and Cryptology”. The second author was supported by a grant of Turkish Academy of Sciences. 465 ¨ ˘ ˙ AKYILDIZ, GULO GLU, IKEDA 2. Set of Characteristics First we fix some notations and conventions. As usual Zn2 stands for the space consisting of all vectors over Z2 of length n. For convenience’ sake we .
