In this paper, we study algebraic geometry (AG) codes with respect to the Lee metric. We determine a new lower bound on the minimum Lee distance of AG codes. Using the AG codes, we construct non-linear binary codes that can correct deletions and insertions of zero-symbols.

For integersu≥ 2 ands≥ 1, letq0 = 2u, and letCs(q0) be the generalized Zetterberg code of lengthn= qs0 + 1 over the finite field Fq0of characteristic 2. For odd characteristic, the covering radius ofCs(q0) was determined recently, whereas the case of even characteristic remained open. In this paper, we determine the covering radius of generalized Zetterberg codes over finite fields of characteristic 2, thereby solving this open problem. Our approach uses methods from the theory of algebraic curves over finite fields. As an application, we obtain an infinite family of quasi-perfect codes.

Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.

For an integer$s\ge 1$, let${\mathcal {C}}_{s}(q_{0})$be the generalized Zetterberg code of length$q_{0}^{s}+1$over the finite field${\mathbb {F}}_{q_{0}}$of odd characteristic. Recently, Shi et al. determined the covering radius of${\mathcal {C}}_{s}(q_{0})$for$q_{0}^{s} \cancel {\equiv }7 \pmod {8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for$q_{0}^{s} \equiv 7 \pmod {8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length$\frac {q_{0}^{s}+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.

Abstract We consider the minimum weight of codewords in a quasi-cyclic code and characterize the estimate in its most general setup using their concatenated structure. The new bound we derive generalizes the Jensen and Güneri–Özbudak bounds and it holds for the more general class of multilevel concatenated codes.

The b-symbol metric is a generalization of the Hamming metric. Linear codes in the b-symbol metric have been used in the read channel whose outputs consist of b consecutive symbols. The Griesmer bound outperforms the Singleton bound for${\mathbb {F}}_{q}$-linear codes in the Hamming metric, when q is fixed and the length is large enough. This scenario is also applicable in the b-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the b-symbol metric. In this paper, we present the b-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the b-symbol Griesmer bound.

Let ${\mathbb{F}_q}$ denote the finite field of size q and $\mathbb{F}_q^n$ denote the set of n-tuples of elements from ${\mathbb{F}_q}$. A family of k-dimensional subspaces of $\mathbb{F}_q^n$, which forms a partial spread, is called L-almost affinely disjoint (or briefly ${[n,k,L]_q}$-AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family.Polyanskii and Vorobyev introduced almost affinely disjoint (AAD) subspace families for $n = 2k + 1$ in [IEEE ISIT 2019 pp. 360–364] using a different language (by saying "L-nice" instead of " ${[n,k,L]_q}$-AAD") in order to construct some types of primitive batch codes. For this purpose, they made use of Reed-Solomon codes and hence they presented $[n = 2k + 1,k,L = k]$-AAD subspace families of size $\left\lfloor {q/k} \right\rfloor $.The general notion of almost affinely disjoint (AAD) subspace families was later introduced and connections with some problems in coding theory were presented by Liu et al. in [Finite Fields Their Appl. 75 (2021) 101879]. In particular, the authors gave upper and lower bounds for the size of AAD subspace families and provided asymptotically optimal constructions of such families for $k = 1$ and $k = 2$ when L is sufficiently large, where the polynomial growth in q is $n - 2k$.Later on, Otal and Arıkan [Finite Fields Their Appl. 84 (2022) 102099] gave some constructions of large AAD subspace families for $n \geq 3k$, hence improved the lower bound of Liu et al. and presented asymptotically optimal AAD subspace families for $n = $ $3k$ when $L \geq 1$.In this paper, we give a construction of large AAD subspace families of size q for $n = 2k + 1$. We also prove that our construction is asymptotically optimal for $k = 2$ and $k = 3$, and conjecture that our construction is still asymptotically optimal for the remaining cases $k > 3$, where $L = k$. Our construction is basically a generalization of a special case of the construction of Otal and Arıkan. We highlight that our construction improves the lower bound given by Polyanskii and Vorobyev to q from $\left\lfloor {q/k} \right\rfloor $. Also we express that our method makes use of linear algebraic techniques rather than Reed-Solomon codes and finite geometry.

Let$ {\mathbb F}_{q_{0}}$be a finite field of odd characteristic. For an integer$s\ge 1$, let$\mathcal {C}_{s}(q_{0})$be the generalized Zetterberg code of length$q_{0}^{s}+1$over$ {\mathbb F}_{q_{0}}$. If$s$is even, then we prove that the covering radius of$\mathcal {C}_{s}(q_{0})$is 3. Put$q=q_{0}^{s}$. If$s$is odd and$q \not \equiv 7 \mod 8$, then we present an explicit lower bound$N_{1}(q_{0})$so that if$s \ge N_{1}(q_{0})$, then the covering radius of$\mathcal {C}_{s}(q_{0})$is 3. We also show that the covering radius of$\mathcal {C}_{1}(q_{0})$is 2. Moreover we study some cases when$s$is an odd integer with$3 \le s \le N_{1}(q_{0})$and, rather unexpectedly, we present concrete examples with covering radius 2 in that range. We introduce half generalized Zetterberg codes of length$(q_{0}^{s}+1)/2$if$q \equiv 1 \mod 4$. Similarly we introduce twisted half generalized Zetterberg codes of length$(q_{0}^{s}+1)/2$if$q \equiv 3 \mod 4$. We show that the same results hold for the half and twisted half generalized Zetterberg codes.

The Galois ring GR$(4^{\Delta})$is the residue ring$Z_{4}[x]/(h(x))$, where$h(x)$is a basic primitive polynomial of degree$\Delta $over$Z_{4}$. For any odd$\Delta $larger than 1, we construct a partition of GR$(4^{\Delta}) \backslash \{0\}$into 6-subsets of type$\{a,b,-a-b,-a,-b,a+b\}$and 3-subsets of type$\{c,-c,2c\}$such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^{\Delta})$and, if$\Delta $is not a multiple of 3, under the action of the automorphism group of GR$(4^{\Delta})$. As a corollary, this implies the existence of quasi-cyclic additive 1-perfect codes of index$(2^{\Delta} -1)$in$D((2^{\Delta} -1)(2^{\Delta} -2)/{6}, 2^{\Delta} -1)$where$D(m,n)$is the Doob metric scheme on$Z^{2m+n}$.

Linear codes with few weights have applications in consumer electronics, communications, data storage systems, secret sharing, authentication codes, and association schemes. As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure computation of data between two parties. The construction of minimal linear codes with new and desirable parameters is an interesting research topic in coding theory and cryptography. Recently, Mesnager et.al. stated that “constructing linear codes with good parameters from non-weakly regular bent functions is an interesting problem.” The goal of this paper is to construct linear codes with two or three weights from non-weakly regular bent functions over finite fields and analyze the minimality of the constructed linear codes. In doing so, we draw inspiration from a paper by Mesnager in which she constructed linear codes with small weights from weakly regular bent functions based on a generic construction method. First, we recall the definitions of the subsets$B_{+}(f)$and$B_{-}(f)$associated with a non-weakly regular bent function$f$. Next, we construct two- or three-weight linear$p$-ary codes on these sets using duals of the non-weakly regular bent functions that are also bent. We note that the constructed linear codes are minimal in almost all cases. Moreover, when$f$is a non-weakly regular bent function in a certain subclass of Generalized Maiorana-McFarland bent functions, we determine the weight distributions of the corresponding linear codes. As far as we know, the construction of linear codes from non-weakly regular bent functions over finite fields is first studied in the literature by the second author in his dissertation.

We prove that the covering radius of the Melas code$M(m,q)$of length$n=q^{m}-1$over$\mathbb {F}_{q}$is 2 if$q > 3$. We also prove that the covering radius of$M(m,3)$is 3 is$m \ge 3$, the covering radius of$M(2,3)$is 4, and the covering radii of$M(1,2)$and$M(1,3)$are 1.

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and \mathbb Fqis an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n-1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q.

Abstract Duan and Lai introduced the notion of “fast point” for a Boolean function f as being a direction a so that the algebraic degree of the derivative of f in direction a is strictly lower than the expected $$\deg (f)-1$$ deg ( f ) - 1 . Their study was motivated by the fact that the existence of fast points makes many cryptographic differential attacks (such as the cube and AIDA attack) more efficient. The number of functions with fast points was determined by Duan et al. in some special cases and by Sălăgean and Mandache-Sălăgean in the general case. We generalise the notion of fast point, defining a fast point of order $$\ell $$ ℓ as being a fast point a so that the degree of the derivative of f in direction a is lower by at least $$\ell $$ ℓ than the expected degree. We determine an explicit formula for the number of functions of degree d in n variables which have fast points of order $$\ell $$ ℓ . Furthermore, we determine the number of functions of degree d in n variables which have a given number of fast points of order $$\ell $$ ℓ , and also the number of functions which have a given profile in terms of the number of fast points of each order. We apply our results to compute the probability of a function to have fast points of order $$\ell $$ ℓ . We also compute the number of functions which admit linear structures (i.e. their derivative in a certain direction is constant); such functions have a long history of being used in the analysis of symmetric ciphers.

Construction of Boolean functions on an odd number of variables with nonlinearity exceeding the bent concatenation bound is one of the most difficult combinatorial problems within the domain of Boolean functions. This problem also has deep implications in coding theory and cryptology. Patterson and Wiedemann demonstrated instances of such functions back in 1983. For more than three decades efforts have been channeled into obtaining such instances. For the first time, in this paper we explore nontrivial upper bounds on nonlinearity for such classes of functions that are invariant not only under several group actions but also for larger sets of functions than what have been considered so far. Further, we present tight upper bounds on the nonlinearity in several cases. To support our claims, we present computational results for functions on $n$ variables, where $n$ is an odd composite integer in the interval [9, 39]. In particular, our results for $n = 15$ and 21 are of immediate interest given recent research results in this domain. In addition to the upper bounds, we also discover the nonlinearities that can actually be achieved above the bent concatenation bound for such a class of functions. Finally, we obtain all possible values in the absolute Walsh spectra of the functions considered.

The design of code division multiple access sequence families dates back to the Gold sequences from the 1960s. Since then there has been a number of different such designs with good correlation properties, some optimal and some near-optimal. In this paper, we use the concept of plateaued functions with arbitrary degree, in order to compute their full correlation distributions. First, we give an explicit correlation distribution of a sequence family using a non-quadratic function. Then for the quadratic functions, we present a general classification of “Gold-like” sequence families for all possible characteristics p and degrees n of the Galois field Fpnused to define the sequences. We are able to obtain the full correlation distribution of the families we consider. This paper also uses techniques from the theory of algebraic curves in order to obtain some of the results.

We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasicyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when C and D are complementary and constacyclic, the codes C and D⊥are equivalent to each other. Hence, the security parameter min(d(C), d(D⊥)) for LCP of codes is simply determined by one of the codes in this case. The same holds for a special class of quasi-cyclic codes, namely 2D cyclic codes, but not in general for all quasi-cyclic codes, since we have examples of LCP of double circulant codes not satisfying this conclusion for the security parameter. We present examples of binary LCP of quasi-cyclic codes and obtain several codes with better parameters than known binary LCD codes. Finally, a linear programming bound is obtained for binary LCP of codes and a table of values from this bound is presented in the case d(C) = d(D⊥). This extends the linear programming bound for LCD codes.

The design of CDMA sequence families using quadratic functions dates back to Gold sequences from the 1960s. Since then there have been a number of different such designs with good correlation properties, some optimal and some near-optimal, and the term “Gold-like” is usually used to denote such sequences. In this paper we use the concept of plateaued functions, not necessarily quadratic, in order to classify such sequence families and present some examples in this direction which depend on the characteristic p and degree n of the Galois field Fpnused to define the sequences.

We give an infinite family of maximum rank distance (MRD) codes, which covers properly the largest known linear MRD code family. Our family contains infinite families of non-linear MRD codes, which are the first non-linear examples for most of the parameters. We also give explicit examples and a table that demonstrates the proportion of linear and non-linear families for some small parameters.

In this paper we generalize a previous construction in order to design perfect autocorrelation sequences over the so-called PSK+ constellation defined by Boztaş and Udaya [2]. We give a number theoretic criterion for the existence of the new sequences with perfect autocorrelation, and discuss some preliminary numerical results on their aperiodic correlations and merit factors.

In this paper a new binary sequence family with 2n+ 1 cyclically distinct sequences each having length 2n- 1 is presented for an even integer n. The correlation distribution of the family is fully determined. The family has six-valued correlation distribution and its maximum correlation magnitude equals 1+2n/2+1.

In this study, the authors give a generalisation of special moduli for faster interleaved Montgomery modular multiplication algorithm with simplified pre‐computational phase for GF ( p n ), where p ≥ 2 is a prime number and n is a positive integer. The authors propose different sets of moduli that can be used in elliptic curve crytographic applications and pairing‐based cryptography. Moreover, this method also leads to efficient implementations for the elliptic curve parameters given in standards. It is shown that one can obtain efficient Montgomery modular multiplication architecture in view of the number of AND gates and XOR gates by choosing proposed sets of moduli. The authors eliminate final substraction step with proposed sets of moduli. These methods are easy to implement for hardware.

Following Jensen's work from 1985, a quasi-cyclic code can be written as a direct sum of concatenated codes, where the inner codes are minimal cyclic codes and the outer codes are linear codes. We observe that the outer codes are nothing but the constituents of the quasi-cyclic code in the sense of Ling-Solé. This concatenated structure enables us to recover some earlier results on quasi-cyclic codes in a simple way, including one of our recent results which says that a quasi-cyclic code with cyclic constituent codes are 2-D cyclic codes. In fact, we obtain a generalization of this result to multidimensional cyclic codes. The concatenated structure also yields a lower bound on the minimum distance of quasi-cyclic codes, as noted by Jensen, which we call Jensen's bound. We show that a recent lower bound on the minimum distance of quasi-cyclic codes that we obtained is in general better than Jensen's lower bound.

We give a general lower bound for the minimum distance of $q$-ary quasi-cyclic codes of length $m\ell$ and index $\ell$, where $m$ is relatively prime to $q$. The bound involves the minimum distances of constituent codes of length $\ell$ as well as the minimum distances of certain cyclic codes of length $m$ which are related to the fields over which the constituents are defined. We present examples which show that the bound is sharp in many instances. We also compare the performance of our bound against the bounds of Lally and Esmaeili-Yari.

We give an efficient multiplication method for polynomials in Chebyshev form. This multiplication method is different from the previous ones. Theoretically, we show that the number of multiplications is at least as good as Karatsuba-based algorithm. Moreover, using the proposed method, we improve the number of additions slightly. We remark that our method works efficiently for any N and it is easy to implement. To the best of our knowledge, the proposed method has the best multiplication and addition complexity for the N-term polynomial multiplication in Chebyshev form with 3 ≤ N ≤ 13.

We give a modified redundant representation for designing arithmetic circuits with small complexity. Using our modified redundant representation, we improve many of the complexity values significantly. Our method works for any finite field. We also give some applications in cryptography.

A method for polynomial multiplication over finite fields using field extensions and polynomial interpolation is introduced. The proposed method uses polynomial interpolation as Toom-Cook method together with field extensions. Furthermore, the proposed method can be used when Toom-Cook method cannot be applied directly. Explicit formulae improving the previous results in many cases are obtained.

Let n and lscr be positive integers and f(x) be an irreducible polynomial over IF2such that lscrdeg(f(x))lscr. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for polynomial multiplication over IF2. We give improved formulae to multiply polynomials of small degree over IF2. In particular we improve the best known multiplication complexities over IF2in the literature in some cases.

We give a new method in order to obtain Weil-Serre type bounds on the minimum distance of arbitrary cyclic codes over${\BBF}_{p^e}$of length coprime to$p$, where$e \ge 1$is an arbitrary integer. In an earlier paper we obtained Weil-Serre type bounds for such codes only when$e=1$or$e=2$using lengthy explicit factorizations, which seems hopeless to generalize. The new method avoids such explicit factorizations and it produces an effective alternative. Using our method we obtain Weil–Serre type bounds in various cases. By examples we show that our bounds perform very well against Bose–Chaudhuri–Hocquenghem (BCH) bound and they yield the exact minimum distance in some cases.

For a prime power q, let $\alpha_q$ be the standard function in the asymptotic theory of codes, that is, $\alpha_q(\delta)$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance $\delta$ of q-ary codes. In recent years the Tsfasman–Vlăduţ–Zink lower bound on $\alpha_q(\delta)$ was improved by Elkies, Xing, Niederreiter and Özbudak, and Maharaj. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. We also show improved lower bounds on the corresponding function $\alpha_q^{\rm lin}$ for linear codes.

We prove a Weil-Serre type bound on the number of solutions of a class of reducible additive equations over finite fields. Using the trace representation of cyclic codes, this enables us to write a general estimate for the weights of cyclic codes. We extend Wolfmann's weight bound to a larger classes of cyclic codes. In particular, our result is applicable to any cyclic code over Fpand Fp2, where p is an arbitrary prime. Examples indicate that our bound performs very well against the Bose-Chaudhuri-Hocquenghem (BCH) bound and that it yields the exact minimum distance in some cases

This paper explores the applications of a recent bound on some Weil-type exponential sums over Galois rings in the construction of codes and sequences. A family of codes over $\F_p$, mostly nonlinear, of length $p^{m+1}$ and size $p^2 \cdot p^{m ( D - \lfloor D/p^2 \rfloor )}$, where $1 \le D \le p^{m/2}$, is obtained. The bound on this type of exponential sums provides a lower bound for the minimum distance of these codes. Several families of pairwise cyclically distinct p-ary sequences of period $p(p^m-1)$ of low correlation are also constructed. They compare favorably with certain known p-ary sequences of period $p^m -1$. Even in the case $p=2$, one of these families is slightly larger than the family $Q(D)$ in section 8.8 in [T. Helleseth and P. V. Kumar, Handbook of Coding Theory, Vol. 2, North-Holland, 1998, pp. 1765-1853], while they share the same period and the same bound for the maximum nontrivial correlation.

We present an upper bound for Weil-type exponential sums over Galois rings of characteristic p/sup 2/ which improves on the analog of the Weil-Carlitz-Uchiyama bound for Galois rings obtained by Kumar, Helleseth, and Calderbank (1995). A more refined bound, expressed in terms of genera of function fields, and an analog of McEliece's (1971) theorem on the divisibility of the homogeneous weights of codewords in trace codes over Z/sub p//sup 2/, are also derived. These results lead to an improvement on the estimation of the minimum distance of certain trace codes over Z/sub p//sup 2/ and the bounds on the correlation of certain nonlinear p-ary sequences.

We show that the concept of "generalized algebraic geometry codes" which was introduced by Xing, Niederreiter, and Lam (see ibid., vol.45, p.2498-2501, Nov. 1999) gives a natural framework for constructing linear unequal error protection codes.

We discuss some previous constructions of codes from algebraic curves due to Xing, Niederreiter and Lam (see ibid., vol.45, no.7, p.2498-2501, 1999), and we investigate their relations with Goppa's (1981) algebraic-geometric codes.