Generalized Reed-Muller Codes Research Articles

Jacobi polynomials for the first-order generalized Reed–Muller codes

In this paper, we give the Jacobi polynomials for first-order generalized Reed–Muller codes. We show as a corollary the nonexistence of combinatorial 3-designs in these codes.

On the codewords of generalized Reed-Muller codes reaching the fourth weight

Recently, the fourth weight of generalized Reed-Muller codes Rq(a(q−1)+b,m) have been characterized for 3≤b<q+43 and 0≤a≤m−3. In this paper, we study the codewords reaching the fourth weight of generalized Reed-Muller codes. We will determine the fourth weight codewords of generalized Reed-Muller codes of order r=a(q−1)+b with 3≤b<q+43 and 0≤a≤m−3. Also, we characterize the number of fourth weight codewords of Rq(b,m) for 3≤b<q+43.

On the fourth weight of generalized Reed–Muller codes

On the fourth weight of generalized Reed–Muller codes

Linear codes of 2-designs as subcodes of the generalized Reed-Muller codes

This paper is devoted to the affine-invariant ternary codes defined by Hermitian functions. We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes. Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs. Finally, we determine the dimension and develop a lower bound on the minimum distance of the ternary linear codes.

Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed-Muller codes.

Steiner systems $$S(2, 4, \frac{3^m-1}{2})$$ and 2-designs from ternary linear codes of length $$\frac{3^m-1}{2}$$

Coding theory and $t$-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus-Mattson theorem and do not admit $2$-transitive or $2$-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed-Muller codes, we prove that they still hold $2$-designs. We also determine the parameters of these $2$-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and $2$-designs from a special type of ternary projective codes.

New Locally Correctable Codes Based on Projective Reed–Muller Codes

Locally decodable codes and locally correctable codes (LCCs) have several important applications, such as private information retrieval, secure multiparty computation, and circuit lower bounds. Three major parameters are considered in LCCs: query complexity, message length, and codeword length. The most familiar LCCs in the regime of low query complexity are the generalized Reed–Muller (GRM) codes. However, it has not previously been determined whether there exist codes that have shorter codeword lengths than GRM codes with the same query complexity and message length. In this paper, we show that the projective Reed–Muller (PRM) codes are such LCCs for some parameters. The GRM code is specified by the alphabet size $q$ , the number of variables $m$ , and the degree $d$ , where $d\le q-2$ . When $d= q-2$ and $q-1$ is a power of a prime, we prove that there exists a PRM code with shorter codeword length than the GRM code with the same query complexity and message length. We also present for these PRM codes a perfectly smooth local decoder to recover a symbol in a codeword by accessing not more than $q$ symbols at the coordinates of the codeword.

Generalized Hamming weights of affine Cartesian codes

Let F be any field and A1,…,Am be finite subsets of F. We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F[x1,…,xm] can have in A1×…×Am. In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed–Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well.

The classification of antipodal two-weight linear codes

By a classical result of Bonisoli, the equidistant linear codes over GF(q) are, up to monomial equivalence, just the replications of some q-ary simplex code, possibly with added 0-coordinates. We prove an analogous result for the antipodal two-weight linear codes over GF(q) (that is, one of the two weights is the length of the code): up to monomial equivalence, any such code is ether a replication of a first order generalized Reed–Muller code, or a replication of a 3-dimensional projective code associated with some non-trivial maximal arc in the classical projective plane PG(2,2s), s≥1.An equivalent geometric formulation of this result reads as follows: a multiset S of points spanning Π=PG(m,q), m≥2, for which every hyperplane intersecting S does so in a constant number of points (counted with multiplicity), is either a multiple of a maximal arc in Π (for m=2) or a multiple of the complement of some hyperplane of Π.

On Bonisoli’s theorem and the block codes of Steiner triple systems

A famous result of Bonisoli characterizes the equidistant linear codes over $${\mathrm{GF}}(q)$$ (up to monomial equivalence) as replications of some q-ary simplex code, possibly with added 0-coordinates. We first prove a variation of this theorem which characterizes the replications of first order generalized Reed–Muller codes among the two-weight linear codes. In the second part of this paper, we use Bonisoli’s theorem and our variation to study the linear block codes of Steiner triple systems, which can only be non-trivial in the binary and ternary case. Assmus proved that the block by point incidence matrices of all Steiner triple systems on v points which have the same 2-rank generate equivalent binary codes and gave an explicit description of a generator matrix for such a code. We provide an alternative, considerably simpler, proof for these results by constructing parity check matrices for the binary codes spanned by the incidence matrix of a Steiner triple system instead, and we also obtain analogues for the ternary case. Moreover, we give simple alternative coding theoretical proofs for the lower bounds of Doyen, Hubaut and Vandensavel on the 2- and 3-ranks of Steiner triple systems.

On monomial codes in modular group algebras

Let p be a prime number and K be the finite field of p elements, i.e. K=GF(p). Further let G be an elementary abelian p-group of order pm. Then the group algebra K[G] is modular. We consider K[G] as an ambient space and the ideals of K[G] as linear codes. A basis of a linear space is called visible, if there exists a member of the basis with the minimum (Hamming) weight of the space. The group algebra approach enables us to find some linear codes with a visible basis in the Jacobson radical of K[G]. These codes can be generated by “monomials” (Drensky & Lakatos, 1989). For p>2, some of our monomial codes have better parameters than the Generalized Reed–Muller codes. In the last part of the paper we determine the automorphism groups of some of the introduced codes.

Identifying Generalized Reed-Muller Codewords by Quantum Queries

We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qn over the finite field of size q. When d is constant, the algorithm needs 𝒪(nd-1) quantum queries, which is optimal. Classically, Ω(nd) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.

The Generalized Reed-Muller Codes in a Modular Group Algebra

First we study some properties of the modular group algebra $\mathbb{F}_{p^r}[G]$ where $G$ is the additive group of a Galois ring of characteristic $p^r$ and $\mathbb{F}_{p^r}$ is the field of $p^r$ elements. Secondly a description of the Generalized Reed-Muller codes over $\mathbb{F}_{p^r}$ in $\mathbb{F}_{p^r}[G]$ is presented.

Hamming distances from a function to all codewords of a Generalized Reed-Muller code of order one

For any finite field \({\mathbb {F}}_q\) with q elements, we study the set \({\mathscr {F}}_{(q,m)}\) of functions from \({\mathbb {F}}_q^m\) into \({\mathbb {F}}_q\) from geometric, analytic and algorithmic points of view. We determine a linear system of \(q^{m+1}\) equations and \(q^{m+1}\) unknowns, which has for unique solution the Hamming distances of a function in \({\mathscr {F}}_{(q,m)}\) to all the affine functions. Moreover, we introduce a Fourier-like transform which allows us to compute all these distances at a cost \(O(mq^m)\) and which would be useful for further problems.

Improved partial permutation decoding for Reed–Muller codes

It is shown that for n≥5 and r≤n−12, if an (n,M,2r+1) binary code exists, then the rth-order Reed–Muller code R(r,n) has s-PD-sets of the minimum size s+1 for 1≤s≤M−1, and these PD-sets correspond to sets of translations of the vector space F2n. In addition, for the first order Reed–Muller code R(1,n), s-PD-sets of size s+1 are constructed for s up to the bound ⌊2nn+1⌋−1. The results apply also to generalized Reed–Muller codes.

On the next-to-minimal weight of affine cartesian codes

In this paper we determine many values of the second least weight of codewords, also known as the next-to-minimal Hamming weight, for a type of affine variety codes, obtained by evaluating polynomials of degree up to d on the points of a cartesian product of n subsets of a finite field Fq. Such codes firstly appeared in a work by O. Geil and C. Thomsen (see [12]) as a special case of the so-called weighted Reed–Muller codes, and later appeared independently in a work by H. López, C. Rentería-Marquez and R. Villarreal (see [16]) named as affine cartesian codes. Our work extends, to affine cartesian codes, the results obtained by Rolland in [17] for generalized Reed–Muller codes.

The Generalized Reed-Muller Codes and the Radical Powers of a Modular Algebra

First, a new proof of Berman and Charpin's characterization of the Reed-Muller codes over the binary field or over an arbitrary prime field is presented. These codes are considered as the powers of the radical of a modular algebra. Secondly, the same method is used for the study of the Generalized Reed-Muller codes over a non prime field.

On the third weight of generalized Reed–Muller codes

In this paper, we study the third weight of generalized Reed–Muller codes. Using results from Erickson, we prove under some condition that the third weight of generalized Reed–Muller codes depends on the third weight of generalized Reed–Muller codes of small order with two variables. We are able to determine the third weight and the third weight codewords of generalized Reed–Muller codes of order r=a(q−1)+b with 2≤b≤q−1, for small b.

Erratum to: On the second weight of generalized Reed-Muller codes

Erratum to: On the second weight of generalized Reed-Muller codes

On low weight codewords of generalized affine and projective Reed–Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed---Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given.