Research On Permutation Decoding Algorithms Of Z₂Z₄-linear Codes

In this paper, we focus on Z2Z4-linear codes. On one hand, we present a new information set which can be used to prove that Z2Z4-linear codes are systematic codes. On the other hand, we correct the application scope of the control condition given in [16] and give a modified result. We can use this result to control the permutation decoding algorithms of Z2Z4-linear codes.For any Z2Z4-linear code, we will present information sets which are different from the ones in [16], and there is a one-to-one corresponding between our informa-tion sets and the ones in [16].For the permutation decoding algorithms of Z2Z4-linear codes, two control con-ditions were given in [16]. One of them is:Let C be a Z2Z4-linear code. Under the Gray map, the preimage of C is an additive code C=Φ-1(C). Suppose the type of C is (α,β;γ,δ;κ) and γ=κ. Under the condition wtH(e)≤t, we have wtL(HΦ-1(y)T)=wtL(HΦ-1^(e)T)≤t(?)wtH(eJ)=0. In this paper, we prove that this control condition is not true for all the additive codes with γ=κ. So we should make some improvements. After these improvements, the control condition could be used for our paper. By using this control condition and the information sets given in this paper, we present permutation decoding algorithms for some Z2Z4-linear codes.At the end of this paper, by using the other control condition given in [16] and the information sets given in this paper, we can present permutation decoding algorithms for all Z2Z4-linear codes.