Constacyclic codes of length $$(p^r,p^s)$$ over mixed alphabets

Multi-dimensional Constacyclic Codes of Arbitrary Length over Finite Fields

“…Assume that q ≡ 1 (mod r ′ ℓ) so that θ ∈ F q . We define χ j (y), the primitive central idempotents in F q [y]/ y r ′ ℓ − 1 similar to ξ t (z) as defined in (5) and then define primitive central idempotents η j (y) in F q [y]/ y ℓ − β as in (7). So we have…”

“…Constacyclic codes over finite fields have a very significant role in the theory of error-correcting codes. A lot of work on constacyclic codes has been done in recent years, see for example [2,4,5,7,15,16]. 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 .…”

“…5 ))= (z − 3)(z + 1)(z + 2). Thus, ζ 0 (z) = −z 2 − 3z − 2, ζ 1 (z) = −2z 2 + 2z − 2, ζ 2 (z) = 3z 2 + z − 2,Also y 2 − 1 = (y − 1)(y + 1), henceη 0 (y) = −3y − 3, η 1 (y) = 3y − 3,We have x 2 − 1 = (x − 1)(x + 1).…”