Twisted group codes of dimension 2 and 3

It is known that all group codes of dimension at most 3 are permutation equivalent to abelian group codes The paper proves that twisted group codes of dimension at most 3 are monomially equivalent either to an abelian group code or to an irreducible constacyclic code

Monomial Equivalence and Code Classification

Monomially equivalent twisted group codes arise naturally in the study of twisted group algebras A rescaling of the basis transforms FαG into another twisted group algebra and every ideal into a monomially equivalent one The natural question is whether every twisted group code of dimension at most 3 is monomially equivalent to an abelian group code Theorem 1 shows that this is not the case, as the ternary tetracode is not monomially equivalent to any group code

Characterization of Twisted Codes

Theorem 2 provides a complete characterization for ideals of dimension at most 3 in twisted group algebras

(a) If α is a coboundary, then I is monomially equivalent to an abelian group code

(b) If α is not a coboundary, then m = dimF I ∈ 2, 3, and there are an α-character ρ: G → L∗ such that I = nXg TrL/F a ρ(g)−1 g: a ∈ L o Moreover, I is monomially equivalent to an irreducible constacyclic code of length G

Constacyclic Codes and Infinite Families

Theorem 3 describes infinite families of irreducible constacyclic codes that attain the Griesmer bound

(i) There is a minimal ideal of F[x]/(x n − c), for some c ∈ F∗, which is a code of length n, dimension k and minimum distance rqk−1

(ii) If k does not divide q − 1, there is also a minimal ideal of F Cn of length n, dimension k and minimum distance rqk−1

(iii) If k divides q − 1 and k does not divide r, then every ideal of a group algebra F G with G = n and dimension k has minimum distance at most rqk−1 − 1

Specific Examples and Limitations

The tetracode is not an isolated example, as it is a particular case of Theorem 3 However, the cocycle of the algebra in (i) is not a coboundary The tetracode is not monomially equivalent to any group code The example 3.10 describes the tetracode explicitly as an ideal in F3[x]/(x4 + 1)

Concluding Remarks

Theorem 2 reduces the study of twisted group codes of dimension at most 3 to two families of codes When the cocycle is a coboundary, the code is monomially equivalent to an abelian group code In any other case, the code is monomially equivalent to an irreducible constacyclic code The second family cannot be avoided as it can be seen in section 4 The tetracode [4, 2, 3]3 can not been monomially equivalent to any group code

Declaration of Competing Interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

Data Availability

No data was used for the research described in the article

Acknowledgments

All the examples in this paper have been verified by computer with assistance from Claude (Anthropic)

Declaration of Generative AI and AI-assisted technologies in the manuscript preparation process

During the preparation of this work the author used Claude (Anthropic) to enhance the writing of the manuscript, check the proofs and compute the examples After using this tool, the author reviewed, verified and edited all mathematical content and takes full responsibility for the content of the published article

References

[1] S.D. Berman, On the theory of group codes, Kibernetika 3 (1) (1967) 31–39 (in Russian); English translation: Cybernetics 3 (1) (1967) 25–31

[2] J.J. Bernal, A. del R´ıo, J.J. Sim´on, An intrinsical description of group codes, Des. Codes Cryptogr. 51 (3) (2009) 289–300

[3] K.S. Brown, Cohomology of Groups, Grad. Texts in Math., vol. 87, Springer, New York, 1982

[4] C. Garc´ıa Pillado, S. Gonz´alez, V. Markov, O. Markova, C. Mart´ınez, Group codes of dimension 2 and 3 are abelian, Finite Fields Appl. 55 (2019) 167–176

[5] C. Garc´ıa Pillado, S. Gonz´alez, V. Markov, C. Mart´ınez, Non-abelian group codes over an arbitrary finite field, Fundam. Prikl. Mat. 20 (2015) 17–22 (in Russian); English translation: J. Math. Sci. 223 (5) (2017) 504–507

[6] J. de la Cruz, W. Willems, Twisted group codes, IEEE Trans. Inform. Theory 67 (2021) 5178–5184

[7] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977

[8] R. Lidl, H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia Math. Appl., vol. 20, Cambridge University Press, 1997

[9] S. Lang, Algebra, 3rd edition, Springer, 2002

[10] G. Karpilovsky, Projective Representations of Finite Groups, Monogr. Textbooks Pure Appl. Math., vol. 94, Marcel Dekker, New York, 1985

[11] I.N. Herstein, Noncommutative Rings, Mathematical Association of America, 1968

[12] W.C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003

[13] B. Schmidt, C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl. 8 (1) (2002) 1–17

[14] A. Behajaina, M. Borello, J. de la Cruz, Twisted skew G-codes, Des. Codes Cryptogr.

1. Monomial equivalence check for twisted group codes can be implemented to classify codes into two main families: abelian group codes (when a cocycle is a coboundary) or irreducible constacyclic codes (when it is not). This allows AI systems to quickly determine the structural class of any given twisted group code.

2. Explicit construction of irreducible constacyclic codes via twisted trace maps provides a method for generating optimal error-correcting codes for specific parameters. AI systems can use this formula to design codes that attain the Griesmer bound, potentially outperforming existing group codes with the same length and dimension.

3. Parameter estimation for twisted group codes up to dimension 3 can be performed by checking if the algebra has a one-dimensional module, which is equivalent to checking if the cocycle is a coboundary. This allows an AI to predict whether an ideal will be monomially equivalent to an abelian or constacyclic code before performing expensive full code analysis.

4. The paper provides tools for comparing codes of length rNk and dimension k by analyzing the set S, which is a subgroup of G related to coordinate maps. AI systems can use this comparison criterion to rapidly assess the minimum distance of a new group code relative to known bounds derived from constacyclic codes.

It is known that all group codes of dimension at most 3 are permutation equivalent to abelian group codes. We prove that twisted group codes of dimension at most 3 are monomially equivalent either to an abelian group code or to an irreducible constacyclic code, and in the second case we describe the code explicitly by means of a trace map. Monomial equivalence cannot be replaced by permutation equivalence, and the second case cannot be avoided: the ternary tetracode is an ideal of F 3[x]/(x 4+1) but it is not monomially equivalent to any group code. More generally, we give two infinite families, in dimensions 2 and 3, of irreducible constacyclic codes which attain the Griesmer bound, whereas every group code with the same length and dimension has smaller minimum distance.