Some distance problems in coding theory

Abstract

Th is thesis is concerned with four topics from coding theory. The first one of these, treated in Chapter 1, is that of coding in an imperfect computer memory with stuck-at-defect3 and random errors. This coding problem finds its crigin in a paper by Kusnetsov and Tsybakov ( 1974). After a short historica! overview in Sectien 1.1, a description of the prOblem and some related problems is given in Sectien 1.2. The Sectiens 1.3 up to 1.5 deal with lower (i.e., constructions) and upper bounds for the various functions defined in Sectien 1.2. The function A(n,d), i.e., the largast size of any binary code of length n and minimum distance d, plays an important rdle in these sections. In Chapter 2 we treat two constructions for constant'weight codes. These constructions result in improved lower bounds on the function Atn,d,w), i.e., the largest size of any binary constant weight code of length n, minimum distance d and constant weight w. This function plays an important role in determining upper bounds ·on the function A (n, d) (e.g.: Linear Programming Sound and Johnson bound). In Chapter 3 we give the complete salution of a problem formulated by Ahlswede, El Gamal and Pang in 1984. They define a constant distance code pair (A,B) as a pair of binary codes of length n such that for some e E N, 0 :> o :> n, They prove that for such a code pair JA • \B . Wi th the help of coding theory Hall and van Lint gave a nice proef of this inequality and moreover characterized all code pairs for which equality holds. Since for these code pairs o = L%J or r%1, the question remained: what happens when o is fixed?. Chapter 3 gives an answer to this question. In Chapter 4 we discuss a problem which arose in conneetion with camma-free codes. Let Wn(q) denote the maximal number of codewordsin any q-ary cernma-free code of length n. Eastman (1965) proved that Wn(q) =.!. I: ll(d)qn/d=: B (q) if n is odd, n dln n For even wordlength n the situation is much more complicated. In 1984 Golomb and Tang proved that where t(kl is the maximal cardinality of any {0,1,*} tournament code of length k. Chapter 4 deals with the problem of determining lower and upper bounds on t(k), k E :ti'. In order to make this thesis self-contained, we start with a short introduetion to coding theo:ry in Chapter 0.