Characterizations of some two-dimensional cyclic codes correspond to the ideals of F[x,y]/<xs−

1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$

“…Özen et al [10] have studied cyclic codes over R. In fact, they determined a generators of the cyclic codes over R. In [10], it was proved that if m is odd, then S is a principal ideal ring. Now, by using a method similar that used for two-dimensional cyclic codes over a field in [13], we obtain a generator polynomials for two-dimensional cyclic codes over R. Our generating set has an important role in determining generator polynomials two-dimensional QC codes over R.…”

“…After that the researchers have studied with different concepts in these codes. The reader can find some of such studies in the papers [11][12][13]. Moreover, Lalason et al [7] construct a basis of an s-dimensional cyclic code over a finite field.…”

Multi-dimensional Constacyclic Codes of Arbitrary Length over Finite Fields

“…In 2016, Sepasdar and Khashyarmanesh [19] obtained generator polynomials of two-dimensional cyclic codes of length s.2 k (s 1 = s, s 2 = 2 k ) over F p m iteratively, where p is an odd prime. The authors state in the concluding remarks of their paper [19] that their method does not work for arbitrary 2-D cyclic codes, not even when s 1 = 3, s 2 = 3.…”

“…In 2016, Sepasdar and Khashyarmanesh [19] obtained generator polynomials of two-dimensional cyclic codes of length s.2 k (s 1 = s, s 2 = 2 k ) over F p m iteratively, where p is an odd prime. The authors state in the concluding remarks of their paper [19] that their method does not work for arbitrary 2-D cyclic codes, not even when s 1 = 3, s 2 = 3. In 2017, Sepasdar [20](unpublished) gave a method for obtaining generator matrix of 2-D cyclic codes of arbitrary length s 1 s 2 , but this construction does not help much in yielding numerical examples.…”

“…Given nonzero elements α and β of F q , a two-dimensional (α, β)-constacyclic code of length sℓ is an ideal of the polynomial ring F q [x, y]/ x s − α, y ℓ − β . In 2018, Rajabi and Khashyarmanesh [14] investigated some repeated-root two-dimensional (α, β)-constacyclic codes of length 2p s .2 k over F p m , where p is an odd prime, using the structure of 2-D cyclic codes given in [19].…”