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Cyclic codes play a very important role in the history of coding theory since they have good algebraic structures that can be widely used in coding and decoding. However, generators for repeated-root cyclic codes of arbitrary length over  F p m u / u k are not unique in previous works for  k ≥ 3 , and hence it is impossible to determine their dual codes. In this work, we propose unique generators for cyclic codes of arbitrary length over  F p m u / u 3 . As its applications, we derive the numbers of their codewords, as well as generators for their dual codes. Furthermore, we propose necessary and sufficient conditions for their self-dualities.

journal of mathematics

Unique Generators for Cyclic Codes of Arbitrary Length over <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mrow> <msup> <mrow> <mi>p</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msup> </mrow> </msub> <mfenced open="[" close="]" separators="|"> <mrow> <mi>u</mi> </mrow> </mfenced> <mo>/</mo> <mrow> <mfenced open="〈" close="〉" separators="|"> <mrow> <msup> <mrow> <mi>u</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </…

Unique Generators for Cyclic Codes of Arbitrary Length over F p m u / u 3

Unique Generators for Cyclic Codes of Arbitrary Length over $F_{p^{m}} [u] / 〈u^{3}〉$