Multi-twisted codes over finite fields and their generalizations

Abstract

Nowadays error-correcting codes are widely used in communication systems, returning pictures from deep space, designing registration numbers, and storage of data in memory systems. An important family of error-correcting codes is that of linear codes, which contain many well-known codes such as Hamming codes, Hadamard codes, cyclic codes and quasi-cyclic codes. Recently, Aydin and Halilovic [5] introduced and studied multi-twisted (MT) codes over the finite field Fq, whose block lengths are coprime to q: These codes are generalizations of well-known classes of linear codes, such as constacyclic codes and generalized quasi-cyclic codes, having rich algebraic structures and containing record-breaker codes. In the same work, they obtained subcodes of MT codes with best-known parameters [33, 12,12] over F3, [53, 18, 21] over F5, [23, 7, 13] over F7 and optimal parameters [54, 4, 44] over F7, Apart from this, they proved that the code parameters [53, 18, 21] over F5 and [33; 12; 12] over F3 can not be attained by constacyclic and quasi-cyclic codes, which suggests that this larger class of MT codes is more promising to find codes with better parameters than the current best known linear codes. In this thesis, we _rst investigate algebraic structures of MT codes over Fq, whose block lengths are coprime to q: We also study their dual codes with respect to Euclidean and Hermitian inner products, and derive necessary and sufficient conditions for a MT code to be (i) self-dual, (iii) self-orthogonal and (iii) linear with complementary-dual (LCD). Applying these results, we provide enumeration formulae for all Euclidean and Hermitian self-dual, self-orthogonal and LCD MT codes over Fq: We also derive some su_cient conditions under which a MT code is either Euclidean LCD or Hermitian LCD. We further develop generator theory for these codes and determine their parity-check polynomials. We also obtain a BCH type bound on their minimum Hamming distances, and express generating sets of Euclidean and Hermitian dual codes of some MT codes in terms of their generating sets. Besides this, we provide a trace description for all MT codes by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Besides this, we explicitly determine all non-zero Hamming weights of codewords of several classes of MT codes over Fq: Using these results, we explicitly determine Hamming weight distributions of several classes of MT codes with a few weights. Among these classes of MT codes with a few weights, we identify two classes of optimal equidistant MT codes that attain the Griesmer as well as Plotkin bounds, and several other classes of MT codes that are useful in constructing secret sharing schemes with nice access structures. We further extend the family of MT codes and study algebraic structures of MT codes over Fq; whose block lengths are arbitrary positive integers, not necessarily coprime to q, We study their dual codes with respect to the Galois inner product and derive necessary and sufficient conditions under which a MT code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD. We also provide a trace description for all MT codes over finite fields by using the generalized discrete Fourier transform (GDFT), which gives rise to a method to construct these codes. We further provide necessary and sufficient conditions under which a Euclidean selfdual MT code over a finite field of even characteristic is a Type II code. We also show that each MT code has a unique normalized generating set. With the help of a normalized generating set, we explicitly determine the dimension and the corresponding generating set of the Galois dual code of each MT code. Besides this, we identify several classes of MT codes over finite fields with a few weights and explicitly determine their Hamming weight distributions. We next study skew analogues of MT codes over finite fields, viz. skew multitwisted (MT) codes, which are linear codes and are generalizations of MT codes. We thoroughly investigate algebraic structures of skew MT codes over finite fields and their Galois duals. Besides this, we view skew MT codes as direct sums of certain concatenated codes and provide a method to construct these codes. We also develop generator theory for these codes, and obtain two lower bounds on their minimum Hamming distances. Finally, we apply our results to obtain many linear codes with best known and optimal parameters from MT and skew MT codes over finite fields.