Let nq(k, d) denote the smallest value ofn for which there exists a linear In, k, d]-code over GF(q). An In, k, d]-code whose length is equal to nq(k, d) is called optimal. The problem of finding nq(k, d) has received much attention for the case q = 2. We generalize several results to the case of an arbitrary prime power q as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite fidd.In particular, we study the problem with q = 3 and determine n3(k, d) for all d when k -< 4, and n3(5, d) for all but 30 values of d.
Tens of thousands of samples of water have been taken from streams, springs, and wells for chemical analysis, without much use having been made of the results. Unfortunately, the chemists have been little concerned with any engineering interpretation of such analyses and the engineers have had no ready method of interpreting them. Consequently, engineers and agriculturists have usually limited themselves to consideration of only the total dissolved solids, or the total hardness, or possibly the amount of chlorides; the real significance of the analyses has been overlooked.
The packing problem for (k, 3)-caps is that of finding (m, 3)r, q, the largest size of (k, 3)-cap in the Galois space Sr, q. The problem is tackled by exploiting the interplay of finite geometries with error-correcting codes. An improved general upper bound on (m, 3)3 q and the actual value of (m, 3)3, 4 are obtained. In terms of coding theory, the methods make a useful contribution to the difficult task of establishing the existence or non-existence of linear codes with certain weight distributions.
Quasi-cyclic codes have provided a rich source of good linear codes. Previous constructions of quasicyclic codes have been confined mainly to codes whose length is a multiple of the dimension. In this paper it is shown how searches may be extended to codes whose length is a multiple of some integer which is greater than the dimension. The particular case of 5-dimensional codes over GF (3) is considered and a number of optimal codes (i.e., [n, k, d]-codes having largest possible minimum distance d for given length n and dimension k) are constructed. These include ternary codes with parameters [45,5, 28], [36,5, 22], [42,5, 26], [48,5, 30] and [72,5, 46], all of which improve on the previously best known bounds.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.