Constant weight codes: An approach based on Knuth's balancing method

Abstract

In this article, we study properties and algorithms for constructing sets of 'constant weight' codewords with bipolar symbols, where the sum of the symbols is a constant q, q = 0. We show various code constructions that extend Knuth's balancing vector scheme, q = 0, to the case where q > 0. We compute the redundancy of the new coding methods.

Construction of q-ary constant weight sequences using a knuth-like approach

“…In the previous section, the construction of q-ary CW sequences was achieved with a weight range shown in (2). However, because of the limited interval, we will present an approach to extend this range.…”

“…In general, generating (n, k, W, q) CW sequence from a q-ary information sequence of length k with the redundant vector u of length e will lead to an increase of weight range. Combining (2) and w(u) ∈ [0, e(q − 1)] results in a CW sequence c = [u|g|y] of weight W is such that…”

“…Example 4: Consider the same ternary information sequence x = 212 of length 3 as in Example 3. We would like to generate a (7, 3, 12, 3) CW sequence of weight W = 12 and n = 7 as described in Table II. We observe from (2) that the information sequence 212 can only generate (n, k, W, q) CW sequences with W ∈ [2,10]. In order to extend this range of weight, we append a ternary redundant vector u of length e to c .…”

“…However, the construction is limited to specific constant dimension codes. In [2], a construction of CW codes based on Knuth's balancing approach [3] is presented. The flexibility on the tail bits is used to generate CW codes including balanced codes.…”

The structure of one weight linear and cyclic codes over Z_{2}^r x (Z_{2} + uZ_{2})^s

“…In [9], Carlet determined one weight linear codes over Z 4 and in [23], Wood studied linear one weight codes over Z m . Constant weight codes are very useful in a variety of applications such as data storage, fault-tolerant circuit design and computing, pattern generation for circuit testing, identification coding, and optical overlay networks [20]. Moreover, the reader can find the other applications of constant weight codes; determining the zero error decision feedback capacity of discrete memoryless channels in [21], multiple access communications and spherical codes for modulation in [14,15], DNA codes in [18,19], powerline communications and frequency hopping in [11].…”