There are many ways to find lower bounds for the minimum distance of a cyclic code, based on investigation of the defining set. Some new theorems are derived. These and earlier techniques are applied to find lower bounds for the minimum distance of ternary cyclic codes. Furthermore, the exact minimum distance of ternary cyclic codes of length less than 40 is computed numerically. A table is given containing all ternary cyclic codes of length less than 40 and having a minimum distance exceeding the BCH bound. It seems that almost all lower bounds are equal to the minimum distance. Especially shifting, which is also done by computer, seems to be very powerful. For length 40>

In the theory of cyclic codes, it is common practice to require that (n,q)=1, where n is the word length and F/sub q/ is the alphabet. It is shown that the even weight subcodes of the shortened binary Hamming codes form a sequence of repeated-root cyclic codes that are optimal. In nearly all other cases, one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime. This statement is based on an analysis for lengths up to 100. A theorem shows why this was to be expected, but it also leads to low-complexity decoding methods. This is an advantage, especially for the codes that are not much worse than corresponding codes of odd length. It is demonstrated that a binary cyclic code of length 2n (n odd) can be obtained from two cyclic codes of length n by the well-known mod u mod u+v mod construction. This leads to an infinite sequence of optimal cyclic codes with distance 4. Furthermore, it is shown that low-complexity decoding methods can be used for these codes. The structure theorem generalizes to other characteristics and to other lengths. Some comparisons of the methods using earlier examples are given.>

It was recently shown that the so-called Jensen bound is generally weaker than the product method and the shifting method introduced by van Lint and Wilson (1986). It is shown that the minimum distance of the two cyclic codes of length 65 (for which it is known that the product method does not produce the desired result) can be proved using Jensen's method (1985) with some adaptations.>

A conjecture of V.C. Da Rocha (see Electron. Lett., vol.21, no.21, p.949-50, 1985) concerning the minimum distance of a class of combinatorial codes is proved.>

THERE ARE fourteen papers on algebraic and combinatorial coding theory in this special issue. Their subject matter mix attests that there are at least two hot topics currently, namely, covering radius problems (four papers) and algebraic geometric codes (three papers); moreover, several other papers on these two areas are in the pipeline. Of course, there are also a few papers on decoding algorithms.

A few years ago Tsfasman {\em et al.,} using results from algebraic geometry, showed that there is a sequence of codes which are generalizations of Goppa codes and which exceed the Gilbert-Varshamov bound. We show that a similar sequence of codes (in fact, the duals of the previous codes) can be found by generalizing the construction of Reed-Solomon codes. Our approach has the advantage that it uses less complicated concepts from algebraic geometry.

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length< 63(with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of length\geq 63.

We show that binary cyclic codes of lengthn = 2^{m} - 1with generatorg(x)=m_{1}(x)m_{7}(x)have minimum distance< 5unlessm = 5(and possiblym = 11, 13, or 17).

Simple descriptions of Preparata and Goethals codes are provided.

A simple definition of generalized quadratic residue codes, that is, quadratic residue codes of block lengthp^{m}, is given, and an account of many of their properties is presented.

A binary code of length 12 and minimum distance 5 with 32 words was discovered by Nadler in 1961. Systematic descriptions of this code given later use larger codes. This correspondence gives a direct construction of Nadler's code.