Group codes over binary tetrahedral group

1 Introduction

The coding theory originated as an inspiration by a famous paper of Shannon [1] published in 1948. Coding theory deals with the problem of detecting and correcting transmission errors which arise when useful information is communicated through a noisy channel. The theory of error correcting codes pertains to the study of reliable transmission of information through a noisy channel using some encoding and decoding procedures. Furthermore, the study of coding techniques involving some encoding and decoding procedures that are based on algebraic techniques for the detection and correction of errors in the transmitted information is called algebraic coding theory.

In the algebraic coding theory, codes constructed using group algebras over Abelian and non-Abelian groups of finite order (see [2,3, 4,5]) are turned out to be ideals or subspaces in the respective group algebras. However, a group algebra code may not form an ideal in some algebraic structures, for instance, the algebraic codes constructed from zero divisors and units in group rings [6] do not form ideals. In addition, group algebra codes play a significant role in the error correction and detection for reliable communications. The development in the study of group algebraic codes got popularized after 1967, when Berman [7] proved that cyclic codes and Reed Muller codes are ideals in a group algebra. Denis Wong and Ang [8] constructed dihedral group codes generated by linear and nonlinear idempotents in a group algebra of dihedral groups of order 6, 8, 10, and 12. Furthermore, in ref. [9] these results were extended to the general case by constructing dihedral group codes generated by linear idempotents in the group algebra of the dihedral group of order 2 n , for arbitrary n . Also, they have obtained dihedral group codes generated by nonlinear idempotents in group algebras of dihedral groups of order 16, 20, and 24.

An important algebraic structure constructed by combining a finite group G and a field K is the group algebra K [ G ] = ∑ g ∈ G a g g : a g ∈ K , which is the free K -module over G with the elements of group G as a K -basis for K [ G ] (see ref. [10]). The set K [ G ] contains all the formal K -linear combinations of elements of group G and it can be made into an associative K -algebra by naturally extending the operations of both K and G to define addition and multiplication of elements in K [ G ] as: ∑ g ∈ G a g g + ∑ g ∈ G b g g = ∑ g ∈ G ( a g + b g ) g ; ∑ g ∈ G a g g ∑ h ∈ G b h h = ∑ x ∈ G ∑ g h = x ∈ G a g b h x , with the identity element 1 = 1 K 1 G , where 1 K and 1 G are the identity elements of K and G , respectively. This associative K -algebra K [ G ] is equipped with the scalar multiplication defined by α ∑ g ∈ G a g g = ∑ g ∈ G ( α a g ) g , for α ∈ K . Moreover, an element g ∈ G can be identified with an element 1 . g in K [ G ] , in this way, G ⊂ K [ G ] and the elements of G constitute the coding basis for group codes when viewed as subspaces in K [ G ] .

A group algebra code in K [ G ] is a one-sided (left or right) ideal in K [ G ] , while if K [ G ] is regarded as a vector space, then a group algebra code is a subspace in K [ G ] . Furthermore, a code in a group algebra K [ G ] is called a cyclic code or an Abelian code, if the group G is cyclic or Abelian. Also, if a code is a vector subspace of K [ G ] , then it is called a linear code. In addition, if char ( K ) ∤ ∣ G ∣ , then K [ G ] is semisimple and hence can be decomposed as a direct sum K [ G ] = ⊕ i K G e k , where K G e k is the minimal ideal generated by the idempotent e k (see ref. [11]). Assume that E = { e i } i = 1 s is the set of idempotents in K [ G ] and if K [ G ] is semisimple, then any ideal I in K [ G ] can be expressed as a direct sum of minimal ideals in { K G e j } e j ∈ E , which are generated by a single element e j ∈ E , i.e., I = ⊕ x ∈ E 1 K G x , for some subset E 1 of E . In this case, an ideal I generated by a subset E ′ = { e i } i = 1 t of the set E in K [ G ] is given by I = { u ∈ K [ G ] : u e j = 0 for all e j ∈ E ⧹ E ′ } and for simplicity I is denoted by I μ , where μ = E ⧹ E ′ , the complement of E ′ in E . Furthermore, the length of a group code I μ in K [ G ] is defined as the order of the group G and is denoted by n . The weight w t ( u ) of an element u = ∑ g ∈ G a g g is the number of nonzero coefficients in the presentation of u with elements of G as basis, i.e., w t ( u ) = ∣ { a g : a g ≠ 0 } ∣ . Let I μ be a group algebra code in K [ G ] with its dimension as a subspace over K is k and the minimum distance d , where d = d ( I μ ) = min { w t ( u ) : 0 ≠ u and u e i = 0 for all e i ∈ μ } . Then the group algebra code I μ is called a ( n , k , d ) group code. Moreover, a linear ( n , k , d ) group code over K is called a maximum distance separable (MDS) group code, if d = n − k + 1 .

Most importantly, the choice of a field K and a group G has a significant influence on the study of group codes in the group algebra K [ G ] . Throughout this article, T is the binary tetrahedral group and K is a splitting field of the group T with char ( K ) ∤ ∣ T ∣ , more precisely, char ( K ) ≠ 2 , 3 . The group T is the semi direct product of the unit quaternion group and the cyclic group of order 3, i.e., T = Q 8 ⋊ Z 3 . This group can also be seen as the preimage of the group of symmetries of tetrahedron under the 2:1 covering homomorphism SU ( 2 ) ⟶ SO ( 3 ) between the special orthogonal group by the spin group, as the group of symmetries of tetrahedron is isomorphic to a subgroup in the special orthogonal group generated by ι 0 0 − ι  and   1 + ι − 1 + ι 1 + ι 0 1 − ι . The group T also has connection with Hurwitz numbers, as the set of invertible Hurwitz numbers ± 1 , ± ι , ± j , ± k , ± 1 ± ι ± j ± k 2 form a group under multiplication, which is isomorphic to the binary tetrahedral group. The group T of order 24 has its abstract presentation as a group generated by r , s , and t satisfying the relations r 2 = s 3 = t 3 = r s t with o ( r ) = 4 , o ( s ) = o ( t ) = 6 .

The main objective of this article is to obtain group codes generated by linear and nonlinear idempotents in the group algebra K [ T ] . To achieve this aim we first compute the idempotents in K [ T ] corresponding to the characters in T . Finally, we establish the group codes in K [ T ] , which are generated by the idempotents in K [ T ] .

2 Group codes in K [ T ]

The binary tetrahedral group T can be regarded as a subgroup generated by the permutations a = ( 4 5 6 7 ) ( 8 9 10 11 ) , b = ( 4 8 6 10 ) ( 5 11 7 9 ) , and c = ( 1 2 3 ) ( 5 11 10 ) ( 7 9 8 ) in the symmetric group S 11 . Thus, the elements of binary tetrahedral group as words using its generators a , b , and c as letters can be listed in a set as follows:

Furthermore, T has seven conjugacy classes which are given by

Consequently, the group T has seven characters, precisely; three linear characters χ 1 , χ 2 , χ 3 and four nonlinear characters χ 4 , χ 5 , χ 6 , χ 7 , out of which the characters χ 4 , χ 5 , χ 6 are of degree 2 and χ 7 is the character of degree 3 and ω represents a cube root of unity as discussed in Table 1.

As we have gathered all the necessary information required to compute the unique idempotents corresponding to each character of T , we list them in the upcoming two lemmas by using Proposition 14.10 of [12], which states that: If χ is a character of K [ G ] -module, then the unique idempotent corresponding to the character χ is given by e = 1 ∣ G ∣ ∑ g ∈ G χ ( g − 1 ) g . Moreover, an idempotent is termed as linear or nonlinear idempotent, if it corresponds to a linear or nonlinear character, respectively. For convenience sake, throughout the rest of this article we denote the class sum of a conjugacy class C i by C i ¯ and the i th element of the set T described in the beginning of this section by g i . The Cayley table of T is given in Table 2.