Optimal quinary cyclic codes with minimum distance four

Abstract

Cyclic codes are an extremely important subclass of linear codes.They are widely used in the communication systems and data storage systems because they have efficient encoding and decoding algorithm.Until now,how to construct the optimal ternary cyclic codes has received a lot of attention and much progress has been made.However,there is less research about the optimal quinary cyclic codes.Firstly,an efficient method to determine if cyclic codes C(1,e,t)were optimal codes was obtained.Secondly,based on the proposed method,when the equation e=5k+1 or e=5m−2hold,the theorem that the cyclic codes C(1,e,t)were optimal quinary cyclic codes was proved.In addition,perfect nonlinear monomials were used to construct optimal quinary cyclic codes with parameters[5m−1,5m−2m−2,4]optimal quinary cyclic codes over F 5 m .

Keywords

• finite field
• cyclic codes
• minimum distance
• perfect nonlinear function