In this paper, we study column twisted Reed-Solomon(TRS) codes. We establish some sufficient conditions for these codes to be MDS and show that the dimension of their Schur square codes is 2k. Consequently, these TRS codes are shown to be not equivalent to Reed-Solomon(RS) codes. Moreover, our construction offers more flexible parameters than existing twisted generalized Reed-Solomon(TGRS) code designs. For a large odd prime powerq, systematically constructed TGRS codes are known to be limited to lengthq+1/2 . By contrast, our column TRS construction supports code lengths up toq+3/2 . Finally, we present the dual codes of column TRS codes. Overall, this work introduces a new method for constructing MDS codes by appending column vectors to some generator matrix of an RS code.

In this paper, we propose a new method for constructing a class of non-GRS MDS codes. The lengths of these codes can reach up toq+3/2 (for finite fields of odd characteristic) andq+4/2 (for even characteristic), respectively. Owing to their special structure, we can use the Cauchy matrix method to obtain the necessary and sufficient conditions for these codes to be MDS codes and non-GRS MDS codes. Additionally, the inequivalence between these codes and twisted GRS codes is analyzed. Furthermore, we analyze the relationships among several existing classes of codes used for constructing non-GRS MDS codes, propose explicit constructions, and discuss the lengths of non-GRS MDS codes based on these constructions. Finally, we design two efficient algorithms to address two main problems in GRS code research, i.e., determining whether an unknown codeCis a GRS code from its generator matrixG, and recovering the key vectors α andvsuch thatC=GRSn,k(α,v) ifCis indeed a GRS code. A computational complexity comparison of the proposed algorithms (O(nk+n)) with that of the Sidelnikov-Shestakov attack (exceedingO(qk2n+qk3))shows that our methods offer superior computational efficiency.

We introduce MorphMamba, a new deep learning architecture for 3D medical image segmentation that successfully adapts the efficiency of state space models to volumetric data. While standard Mamba models excel at long-sequence modeling, they fail to preserve the rich 3D spatial context of medical images due to a required data-flattening step. MorphMamba overcomes this critical limitation with a novel Mamba-based encoder designed for volumetric inputs. Our architecture integrates two key innovations: a Channel-Spatial Attention Guided module, which refines feature maps to emphasize salient anatomical information before sequential processing, and a Context Refining Block in the skip-connections to ensure optimal multi-scale feature fusion. We demonstrate the state-of-the-art performance of MorphMamba through extensive experiments on four diverse benchmarks, including BraTS 2023, LiTS 2017, a private cardiac dataset and a private prostate tumor dataset. Our model consistently achieves superior results in terms of both accuracy and boundary delineation, establishing a new and efficient baseline for 3D segmentation. Our code is available at https://github.com/haoranGeng/MorphMamba.

In this paper, firstly, we study decoding of a general class of twisted generalized Reed-Solomon (TGRS) codes and provide a precise characterization of the key equation for TGRS codes and propose a decoding algorithm. Secondly, we further study decoding of almost-MDS TGRS codes and provide a decoding algorithm. These two decoding algorithms are more efficient in terms of performance compared with the decoding algorithms presented in [Sun et al., IEEE-TIT, 2024] and [Sui et al., IEEE-TIT, 2023] respectively. Moreover, these two optimized decoding algorithms can be applied to the decoding of a more general class of twisted Goppa codes.

Let$n$be a positive integer and$p$a prime. The power mapping$x^{p^{n}-3}$over${\mathbb {F}}_{p^{n}}$has desirable differential properties, and its differential spectra for$p=2,\,3$have been determined. In this paper, for any odd prime$p$, by investigating certain quadratic character sums and some equations over${\mathbb {F}}_{p^{n}}$, we determine the differential spectrum of$x^{p^{n}-3}$with a unified approach. The obtained result shows that for any given odd prime$p$, the differential spectrum can be expressed explicitly in terms of$n$. Compared with previous results, a special elliptic curve over${\mathbb {F}}_{p}$plays an important role in our computation for the general case$p \ge 5$.

In this paper, two criterions of MDS Euclidean self-orthogonal codes are presented. New MDS Euclidean self-dual codes and self-orthogonal codes are constructed via the criterions. In particular, among our constructions, for large square q, about 1/8·q new MDS Euclidean (almost) self-dual codes over Fqcan be produced. Moreover, we can construct about 1/4· q new MDS Euclidean self-orthogonal codes with different even lengths n and dimension n/2-1.

An error model with asymmetric single error with magnitude four is considered. In this paper, the constructions of codes correcting single error of magnitude four over \mathbb Z2a3brare studied which is equivalent to construct B1[4](2a3br) sets. Firstly, we reduce the construction of a maximal size B1[4](2a3br) set for a ≥ 4 and gcd(r,6)=1 to the construction of a maximal size B1[4](2a-33br) set. Furthermore, we will show that maximal size B1[4](8·3br) sets can be reduced to maximal size B1[4](3br) sets and also give lower bounds of maximal size B1[4](12r) and B1[4](2·3br) sets. Finally, we give a necessary and sufficient condition on the existence of perfect B1[4](p) set for prime p.

Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over Z2krwhich is based on so called B1[4](2kr) set. In fact, we reduce the construction of a maximal size B1[4](2kr) set for k ≥ 3 to the construction of a maximal size B1[4](2k-3r) set. Finally, we give an explicit formula of a maximal size B1[4](4r) set and some lower bounds of a maximal size B1[4](2r) set. By computer searching up to q ≤ 106, we conjecture that those lower bounds are tight.

In general, it is hard to find the cross correlation distribution between two sequences. In particular, there are only few cases that the cross correlation is three-valued. In this paper, new pairs of binary sequences with three cross correlation values are presented. These two sequences have different lengths: 22k- 1 and 3(2k- 1), respectively. We determine the cross correlation distribution between these two sequences. We also give the possible correlation values for the shift decimated sequences. The magnitude of cross correlation values is shown to be low. The techniques presented in this paper may be useful to handle other similar cases.

In this paper, we consider a class of exponential sums from some half quadratic binomials. The exponential sums are proven to be eleven-valued with maximal magnitude (1/2(q - √q)) except for the trivial value q. As applications, first, we investigate the autocorrelation and cross-correlation distribution among the sequences in a sequence family. Second, we determine the weight distributions of several classes of linear codes. Some of the dual codes have minimum distance four, which are optimal with respect to the Hamming bound. Our results extend the result by Choi et al. and show that some correlation values in it do not occur.

In this paper, a class of exponential sum from some binomials is studied. As an application, we determine the weight distribution of a family of p-ary cyclic codes.

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of p-ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.

Let m be odd and q = 22m. Let 5r ≡ 1 (mod 2m+ 1) and d = (2m- 1)r + 1. In this paper, the cross correlation distributions of an m-sequences with period q - 1 and its decimated sequences s(dt + l) with period (q - 1)/3 for 0 ≤ l ≤ 2 are determined. These cross correlations are shown to be four-valued with maximal magnitude 2√q - 1.

Upper bounds on the weight distribution function for codes of minimum distance at least 2 are given. Codes, where the bound is met with equality, are characterized. An improved upper bound on the weight distribution function for codes of minimum distance at least 3 is given. As an application, a sharp upper bound on the probability of undetected error for linear codes with full support is characterized.

An error model with symmetric errors of limited magnitude is considered. Limited magnitude means that the size of any error is limited by a number smaller (usually much smaller) than the alphabet size. Several constructions of codes correcting a single error are given. In some cases, the codes are perfect or quasi- perfect.

An error model with asymmetric errors of limited magnitude is a good model for some multilevel flash memories. This paper is about constructions of codes correcting such errors. The main results are about codes correcting a single such error and codes of lengthmcorrecting all errors inm-1 or less positions.

Constant composition codes are codes where the frequency distribution of the elements in a codeword is the same for all codewords. In this paper, three classes of constant composition codes are constructed. These codes are subcodes of cyclic codes which have few weights occurring among the codewords. The new codes are excellent asymptotically compared to the previously best known constant composition codes.

In this paper the correlation distribution of a p-ary m-sequence s and its decimated sequence s(dt + l) with d = ((pm+ 1)/2)2and 0 ≤ l ≤ d - 1 is determined.

In this paper, we first construct several classes of classical Hermitian self-orthogonal maximum distance separable (MDS) codes. Through these classical codes, we are able to obtain various quantum MDS codes. It turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions.

We generalize Steane’s enlargement construction of binary quantum codes to q-ary quantum codes. We then apply this result to BCH codes and the study of asymptotic bounds, and obtain improvements to the quantum BCH codes constructed by Aly and Klappenecker and the quantum asymptotic bounds from algebraic geometry codes obtained by Feng, Ling and Xing.

In this paper, the large family of generalized Kasami sequences has been studied. In particular, the cross-correlation distribution among these sequences has been explicitly calculated. Meanwhile, the weight distributions of two classes of cyclic codes could also be determined. This paper generalizes the results from several previous papers.

Let q = pnwith n = 2m and p be a prime. Let 0 ≤ k ≤ n − 1, k ≠ m. Assume k/(m,k) and m/(m,k) are both odd. In this paper we will study the following exponential sums of the equation where Tr1n: Fq− Fpand Tr1m: Fpm→ Fpare the canonical trace mappings and ζp= e 2πi/p is a primitive p-th root of unity. As an application, we will determine the weight distribution of the cyclic codes C over Fpwith parity-check polynomial h1(x)h2(x) where h1(x) and h2(x) are the minimal polynomials of πequation and πequation over Fprespectively for a primitive element π of Fq.

Let q=pmwhere p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from Fqto Fpand zetap=e2pii/pbe a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: SigmaxisinFqchi(alphaxpk+1+betax2) (alpha, betaisinFq) where chi(x)=zetapTr(x)is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1and C2over Fpt with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where t is a divisor of d=gcd(m, k), and h1(x), h2(x) , and h3(x) are the minimal polynomials of pi-1, pi-2, and pi-(pk+1)over Fpt, respectively, for a primitive element pi of Fq. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).

In this paper, we will study the exponential sum SigmaxisinFqx(alphax(pk+1)/2+betax) that is related to the generalized Coulter-Matthews function x(pk+1)/2 with k/gcd. As applications, we obtain the following: the correlation distribution of a p-ary m-sequence and a decimated m-sequence of degree pk+1/2; the weight distribution of the cyclic code whose dual has two zeros pi-1and pi-((pk+1)/2).

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums$$\sum _{x\in\BBF _{q}}\zeta _{p}^{{\rm Tr}\left(af(x)+bx\right)}\left(a,b\in \BBF _{q},q=p^{m},p\ge 3\right)$$for all perfect nonlinear functions$f$which is a Dembowski–Ostrom polynomial or$p\!=\!3$,$f\!=\!x^{{ 3^{k} + 1}\over { 2}}$where$k$is odd and$(k,m)\!=\!1.\break$As applications, we determine 1) the correlation distribution of the$m$-sequence$\left \{a_{\lambda }= {\rm Tr}(\gamma ^{\lambda })\right \}({\lambda =0,1,\ldots })$and the sequence$\left \{b_{\lambda }= {\rm Tr}\left (f(\gamma ^{\lambda })\right)\right \}({\lambda =0,1,\ldots })$over$\BBF _{p}$where$\gamma $is a primitive element of$\BBF _{q}$and 2) the weight distributions of the linear codes over$\BBF _{p}$defined by$f$.