In this article, we use group action theory to define some important ternary Linear Codes. Some of these Codes are self-orthogonal having a minimum distance achieving the lower bound in the previous records. Then, we define two new Codes sharing the same automorphism group isomorphic to $C_2 \times M_{11}$ where $M_{11}$ is the Sporadic Mathieu group and $C_{2}$ is a cyclic group of two elements. We also study the natural action of the general Linear group $GL (k, 2) $ on the vector space $F_2 ^ k$ to characterize Hamming Codes $H_k (2) $ and their automorphism group.

Let Zp Zp be the finite field of integers modulo p p , where p>3 p>3 is a prime integer. This paper presents new constructions of Linear Codes over Zp Zp . Based on our construction, Linear Codes of length p− 1 p− 1 , including a wide family of MDS Codes, and Codes of length (p− 1)2 (p− 1)2 are constructed. we shall discuss the parameters of the Codes defined while describing a generator matrix for the first family.

In this paper, we consider the minimum Hamming weight for Linear Codes over special finite quasi-Frobenius rings. Furthermore, we obtain minimal free R-submodules of a finite quasi-Frobenius ring R which contain a Linear code and derive the relation between their minimum Hamming weights. Finally, we suggest an algorithm that computes this weight using the Gr¨ obner basis and we show that under certain conditions a Linear code takes the maximum of minimum Hamming weight.

Steganography is an information hiding application which aims to hide secret messages imperceptibly into commonly used media. In this paper, we describe an Optimal embedding method based on Linear Codes that conforms to least significant bit, that is, the secret data is embedded into a cover message by parity check matrix. The new method not only benefits from the field of location and detection and error correction bit stream received by the receiver, but also can increase the Resistance between 94% to 100%, the transparency (PSNR) up to 84.71, and the similarity (SSIM) up to % 9999.99.

Let nq (k, d) denote the smallest value of n for which there exists a Linear [n, k, d]-code over the Galois field GF (q). An [n, k, d]-code whose length is equal to nq (k, d) is called optimal. In this paper we present some matrix generators for the family of optimal [n, 3, d] Codes over GF (7) and GF (11). Most of our given Codes in GF (7) are non-isomorphic with the Codes presented before. Our given Codes in GF (11) are all new.