Can Xiang

Can Xiang, Chunming Tang

Cyclic maximum distance separable (MDS for short) codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs and finite geometry. However, the existing construction methods for cyclic MDS codes are mainly focused on strict restrictions on certain parameters or are relatively complex in their construction approaches. In this paper, we investigate this approach further via norm reduction in cyclotomic fields and present a construction method of cyclic MDS codes over finite fields. We transform the problem of verifying the MDS property over a finite field into a problem of determining non-zero minors in characteristic zero. Compared with existing construction methods, our method is relatively simple. In particular, the results of this paper show that the parameters of non-RS cyclic MDS codes are flexible and completely cover the results in [Non-Reed-Solomon Type Cyclic MDS codes, IEEE Trans. Inf. Theory, 71(5): 3489--3496, 2025].

Yuehui Cui, Jinquan Luo, Can Xiang

Recently, the Walsh spectrum and boomerang properties of special power functions have aroused widespread research interest, owing to their important applications in cryptography and information security. In particular, locally-APN functions may offer superior resistance against differential cryptanalysis compared to other functions of equivalent differential uniformity. Up till now only a small number of locally-APN functions have been studied. In this paper, we show that a Niho type power function is locally-APN if and only if its Walsh spectrum takes four values in \(\{-p^m, 0, p^m, 2p^m\}\). Equivalently, the associated cyclic codes have four nonzero weights: $p^{m-1}(p-1)(p^m + k)$ for $k = 0, 1, -1, -2$. Moreover, we also study properties of Niho type locally-APN power functions, including their differential spectrum, Walsh spectrum, Feistel Boomerang Connectivity Table(FBCT for short) and second-order zero differential spectra.