In this paper, a new view point to linear codes is presented. A concept level of linear codes is introduced and a bound of linear code’s level is given. It can be used to simplify the construction of Algebraic Geometric Codes. In particular, we define a class of codes which can be considerated as generalized RS codes and can be constructed via symbolic computation.

We introduced error-correcting codes and linear codes in the last lecture. In this lecture we will discuss in some more details properties of linear codes and we’ll describe classical examples of linear codes. We will also show the Hamming bound, which is a bound that relates the distance, rate, and block length parameters of codes and is tight for the Hamming codes. Recall that a (n, k, d)q co...

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of p-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field Fp, where p is an odd prime. They include some earlier linear codes as special cases. The weight distributions...

We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring Zm. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over Z8 and Z9 of lengths up to 6. We determine the minimum distances of optimal linear codes over Z4 for lengths up to 7. Some examples of optimal codes are given.

There are several methods for constructing secret sharing schemes, one of which is based on coding theory. Theoretically, every linear code can be used to construct secret sharing schemes. However, in general, determining the access structures of the schemes based on linear codes is very hard. This paper proposed the concept of minimal linear code, which makes the determination of the access st...

In this paper, for the purposes of information transmission and network error correction simultaneously, three classes of important linear network codes in network coding, linear multicast/broadcast/dispersion codes are generalized to linear network error correction coding, i.e., linear network error correction multicast/broadcast/dispersion codes. We further propose the (weakly, strongly) exte...

With the help of a computer, we obtain the minimum distance of some codes belonging to two families of Z2k -linear codes: the first is the generalized Kerdock codes which aren’t as good as the best linear codes and the second is the Hensel lift of quadratic residue codes. In the latter, we found new codes with same minimum distances as the best linear codes of same length and same cardinality. ...

We further the study of the duality theory of linear space-time codes over finite fields by showing that the only finite linear temporal correlated codes with a duality theory are the column distance codes and the rank codes. We introduce weight enumerators for both these codes and show that they have MacWilliams-type functional equations relating them to the weight enumerators of their duals. ...