In this paper, we use extended Euclid’s algorithm to propose new decoding algorithms for two classes of maximum distance separable (MDS) twisted generalized Reed-Solomon (TGRS) codes of parameters$[n, n-t, t+1]$over$\Bbb F_{q}$. For even t, the algorithms can correct$\frac {t}{2}$errors with time complexity$O(qn)$. Moreover, we also give a new decoding algorithm for a class of twisted Goppa codes. For even degree t of a Goppa polynomial, it can also correct$\frac {t}{2}$errors, which generalizes a$\lfloor \frac {t-1}{2}\rfloor $-error-correcting decoding algorithm by Sui and Yue (2023).

Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let$\Bbb F_{q}$be a finite field with$q=2^{l}$elements, where l is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism$A \in PGL_{2}(\Bbb F_{q})$. Moreover, we provide some examples to support our findings.

In this paper, we introduce twisted Goppa codes, which generalize classical Goppa codes by adding a twisted term. Then we provide an efficient decoding algorithm for twisted Goppa codes. The Niederreiter cryptosystem is bassed on linear error-correcting codes in which the public key is a parity check matrix. When twisted Goppa codes are applied to the Niederreiter cryptosystem, the public key size is overlarge. To reduce the public key size, we construct quasi-cyclic twisted Goppa codes via a non-trivial automorphism group carefully selecting the defining set and the matched polynomial. Moreover, we obtain a family of cyclic twisted Goppa codes.

Let$n (>3)$be a prime number and${\mathbb {F}}_{2^{n}}$a finite field of$2^{n}$elements. Let$L ={\mathbb {F}}_{2^{n}}\cup \{\infty \}$be the support set and$g(x)$an irreducible polynomial of degree 6 over${\mathbb {F}}_{2^{n}}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes$\Gamma (L, g)$of degree 6 and length$2^{n}+1$.

In this paper, we determine all dihedral subgroups in$PGL_{2}(\mathbb {F}_{q})$, where$q=2^{l}$and$l$is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes.

Twisted generalized Reed-Solomon (TGRS) codes are a family of codes that contains a large number of maximum distance separable (MDS) codes that are non-equivalent to generalized Reed-Solomon (GRS) codes. In this paper, we characterize a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code with two twists is MDS; give a sufficient and necessary condition that a TGRS code with two twists is self-dual, and present some constructions of self-dual TGRS codes. These self-dual codes are MDS, NMDS or 2-MDS. Furthermore, we study the non-GRS properties of TGRS codes with two twists and prove that these codes are non-GRS in most cases.

A Goppa code over \Bbb Fqmis a well-known subclass of algebraic error-correcting code. If m=1, then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters.

Let Fqbe a finite field with q elements and p be a prime with gcd(p, q) = 1. Let G be a finite abelian p-group and Fq(G) be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra Fq(G). Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of Fq(G).

Recently, some infinite families of binary minimal and optimal linear codes were constructed from simplicial complexes by Hyun et al. Inspired by their work, we present two new constructions of codes over the ring F2+ uF2by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over F2+ uF2. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes.

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let${\mathbb {F}}_{q}$be a finite field with order$q$, where$q$is a positive power of a prime$p$. Suppose that$n$is a positive integer and the product of distinct prime factors of$n$divides$q-1$, i.e.,$rad(n)\mid (q-1)$. In this paper, we explicitly factorize the polynomial$x^{n}-\lambda $for each$\lambda \in {\mathbb {F}}_{q}^{*}$. As applications, first, we obtain all repeated-root$\lambda $-constacyclic codes and their dual codes of length$np^{s}$over${\mathbb {F}}_{q}$; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length$n$over${\mathbb {F}}_{q}$; third, we list all self-dual repeated-root negacyclic codes of length$np^{s}$over${\mathbb {F}}_{q}$. In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters.

Let$\Bbb F_{p}$be a finite field with$p$elements, where$p$is a prime. Let$N \ge 2$be an integer and$d$be the least positive integer satisfying$p^{d} \equiv -1 \pmod N$. Let$q = p^{2sd}$for some integers$s$. In some special cases, we obtain the explicit evaluation of the following exponential sums:$S(a,b)=\sum _{x\in \Bbb F_{q}^{*}}\zeta _{p}^{ \mathrm {Tr}_{q/p}(ax^{(({q-1})/{N})}+bx)}$. As applications, Walsh spectrums of the monomial functions$\mathrm {Tr}_{q/p}(x^{(({q-1})/{N})})$in three cases are investigated. Our results show that Walsh spectrums of the monomial functions have at most four, five, or seven distinct values. Furthermore, three families of the monomial functions with three-valued Walsh spectrums are presented, seeCorollaries 12,21,31, and32. Consequently, certain previously known results by Li and Yue and Moisio are extended.

In practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing, and codebooks with small inner-product correlation are required. It is extremely difficult to construct codebooks achieving the Levenshtein bound. In this paper, two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented. These constructions produce complex codebooks asymptotically achieving the Levenshtein bound and codebooks asymptotically achieving the Welch bound. The codebooks presented in this paper have new parameters.

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega's result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega.

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight distributions are known only for a few cases. In this paper, let Frbe an extension of a finite field Fqand r = qm, we determine the weight distributions of the cyclic codes C={c(a, b): a, b ∈ Fr}, c(a, b)= Trr/q(ag10+bg20),...,Trr/q(ag1n-1+bg2n-1)), g1, g2∈ Fr, in the following two cases: 1) ord(g1)=n, n|r-1 and g2=1 and 2) ord(g1)=n, g2=g12, ord(g2)=n/2, m=2, and 2(r-1)/n|(q+1).

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let Fr be a finite field with r = qm. Suppose that g1, g2 ∈ F*rare not conjugates over Fq, ord(g1) = n1, ord(g2) = n2, d = gcd(n1, n2), and n = n1n2/d. Let Fq(g1) = Fqm1, Fq(g2) = Fqm2, and Ti denote the trace function from Fqmito Fq for i = 1, 2. We define a cyclic code C(q,m,n1,n2) = {c(a, b) : a ∈ Fqm1, b ∈ Fqm2}, where c(a, b) = (T1(ag01) + T2(bg02), T1(ag11) + T2(bg12), ... , T1(agn-11) + T2(bgn-12)). We mainly use Gauss periods to present the weight distribution of the cyclic code C(q,m,n1,n2). As applications, we determine the weight distribution of cyclic code C(q,m,qm1-1,qm2-1) with gcd(m1, m2) = 1; in particular, it is a three-weight cyclic code if gcd(q -1, m1 -m2) = 1. We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes.

In this paper, we mainly get three results. First, let p, q be distinct primes with ((p-1)p,(q-1)q)=(p-1,q-1)=e ; we give a method to compute the linear complexity of Whiteman's generalized cyclotomic sequences of period pm+1qn+1. Second, if e=4, we compute the exact linear complexity of Whiteman's generalized cyclotomic sequences. Third, if p≡q 5 (mod 8), gcd(p-1, q-1)=4, and we fix a common primitive root g of both p and q, then 2∈H0=(g), which is a subgroup of the multiplicative group Z*pq, if and only if Whiteman's generalized cyclotomic numbers of order 4 depend on the decomposition pq=a2+4b2with 4|b.