Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let   F q        F    q          be a finite field of odd order   q     q        and            n  =    2    a      p    1      a    1        p    2      a    2           , where   a,a 1 ,a 2      a  ,    a    1    ,    a    2          are positive integers,   p 1 ,p 2        p    1    ,    p    2          are distinct odd primes and   4p 1 p 2 |q−1     4    p    1      p    2    |  q  −  1       . ,  study the irreducible factorization of   x n −1       x    n    −  1        over   F q        F    q          and all primitive idempotents in the ring   F q [x]∕〈x n −1〉       F    q     [  x  ]   ∕   〈    x    n    −  1  〉        .Moreover,  obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length   n     n        over   F q        F    q         .
['Fen Li', 'Xiwang Cao']
I am writing a research paper on a idea A class of minimal cyclic codes over finite fields
[{"from": "human", "value": "I am writing a research paper on a idea A class of minimal cyclic codes over finite fields"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let   F q        F    q          be a finite field of odd order   q     q        and            n  =    2    a      p    1      a    1        p    2      a    2           , where   a,a 1 ,a 2      a  ,    a    1    ,    a    2          are positive integers,   p 1 ,p 2        p    1    ,    p    2          are distinct odd primes and   4p 1 p 2 |q\u22121     4    p    1      p    2    |  q  \u2212  1       . ,  study the irreducible factorization of   x n \u22121       x    n    \u2212  1        over   F q        F    q          and all primitive idempotents in the ring   F q [x]\u2215\u3008x n \u22121\u3009       F    q     [  x  ]   \u2215   \u3008    x    n    \u2212  1  \u3009        .Moreover,  obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length   n     n        over   F q        F    q         ."}]