code

We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the alternating group A7. It is shown that a self-dual symmetric 2-(35, 18, 9) design and an optimal even binary [21, 14, 4] LCD code are found such that they are invariant under the full automorphism groups S8 and S7, respectively. Moreover, designs with parameters 1-(21, l, k1,l) and 1-(35, l, k2,l) are obtained, where ω is a codeword, l = wt(ω), k1,l = l|ω S7 |/21 and k2,l = l|ω S7 |/35. It is seen that there exist a 2-(21, 5, 12) design with the full automorphism group S7 among these 1-designs

Let $G=(V, E)$ be a simple graph. A set $C$ of vertices $G$ is an identifying set of $G$ if for every two vertices $x$ and $y$ belong to $V$ the sets $N_{G}[x] \cap C$ and $N_{G}[y] \cap C$ are non-empty and different. Given a graph $G,$ the smallest size of an identifying set of $G$ is called the identifying code number of $G$ and is denoted by $\gamma^{ID}(G).$ Two vertices $x$ and $y$ are twins when $N_{G}[x]=N_{G}[y].$ Graphs with at least two twin vertices are not identifiable graphs. In this paper,  we present three bounds for identifying code number.

In this paper, we use the Key-Moori Method 1 and construct a quaternary code $mathcal{C}_8$ from a primitive representation of the group $PSL_2(9)$ of degree 15. We see that $mathcal{C}_8$ is a self-orthogonal even code with the automorphism group isomorphic to the alternating group $A_8$. It is shown that by taking the support of any codeword $omega$ of weight $l$ in $mathcal{C}_8$ or $mathcal{C}_8^bot$, and orbiting it under $A_8$, a 2-$(15,l,lambda)$ design invariant under the group $A_8$ is obtained, where $lambda=binom{l}{2}|omega^{A_8}|/binom{15}{2}$. A number of these designs have not been known before up to our best knowledge. The structure of the stabilizers $(A_8)_omega$ is determined and moreover, primitivity of $A_8$ on each design is examined.

Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was first introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to find many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G

For any prime p let Cp(G) be the p-ary code spanned by the rows of the incidence matrix G of a graph Γý. ýLet Γ be the incidence graph of a flag-transitive symmetric design Dý. ýWe show that any flag-transitiveý ýautomorphism group of D can be used as a PD-set for full error correction for the linear code Cp(G)ý ý(with any information set)ý. ýIt follows that such codes derived from flag-transitive symmetric designs can beý ýdecoded using permutation decodingý. ýIn that way to each flag-transitive symmetric (vý,ýký,ýλ) design we associate a linear code of length vk that isý ýpermutation decodableý. ýPD-sets obtained in the described way are usually of large cardinalityý. ýBy studying codes arising from some flag-transitive symmetric designs we show that smaller PD-sets can be found forý ýspecific information setsý.

We construct two classes of Gray maps, called type-I Gray map and type-II Gray map, for a finite $p$-group $G$. Type-I Gray maps are constructed based on the existence of a Gray map for a maximal subgroup $H$ of $G$. When $G$ is a semidirect product of two finite $p$-groups $H$ and $K$, both $H$ and $K$ admit Gray maps and the corresponding homomorphism $psi:Hlongrightarrow {rm Aut}(K)$ is compatible with the Gray map of $K$ in a sense which we will explain, we construct type-II Gray maps for $G$. Finally, we consider group codes over the dihedral group $D_8$ of order 8 given by the set of their generators, and derive a representation and an encoding procedure for such codes.