Generalized Reed-Muller Codes Research Articles

Subcodes of the Projective Generalized Reed-Muller Codes Spanned by Minimum-Weight Vectors

We use methods of Mortimer l19r to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length \frac{q^m - 1}{q-1} over the finite field Fq of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0<r<m−1 unless q is prime. The methods of proof are the projective analogue of those developed in l17r, and show that the codes spanned by the minimum-weight vectors are spanned over Fq by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1.

Matrix-Product Codes over ? q

Codes C 1 ,…,C M of length n over ? q and an M × N matrix A over ? q define a matrix-product code C = [C 1 …C M ] ·A consisting of all matrix products [c 1 … c M ] ·A. This generalizes the (u|u+v)-, (u+v+w|2u+v|u)-, (a+x|b+x|a+b+x)-, (u+v|u-v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ⊥. It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be `non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary `Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ⊥ can be described using C 1 ⊥, …,C M ⊥ and a transformation of A. This yields d(C ⊥). Finally we show that an NSC matrix-product code is a generalized concatenated code.

Codes from Veronese and Segre Embeddings and Hamada's Formula

In this article we study the codes given by l hypersurfaces in Pnq to obtain a new formula for the dimension of codes given by (n−l) flats. We also obtain a new formula for the dimension of the ν th order generalized Reed–Muller code and describe the code given by the hyperplane intersections of the Segre embedding of Pnq×Pmq.

The degree of functions and weights in linear codes

Properties of the weight distribution of low-dimensional generalized Reed–Muller codes are used to obtain restrictions on the weight distribution of linear codes over arbitrary fields. These restrictions are used in non-existence proofs for ternary linear code with parameters [74,10,44] [82,6,53] and [96,6,62].

Some Applications of Magma in Designs and Codes: Oval Designs, Hermitian Unitals and Generalized Reed–Muller Codes

We describe three applications of Magma to problems in the area of designs and the associated codes: • Steiner systems, Hadamard designs and symmetric designs arising from an oval in an even-order plane, leading in the classical case to bent functions and difference-set designs; • the Hermitian unital as a 2-(q3+ 1, q+ 1, 1) design, and the code overFp where p divides q+ 1; • a basis of minimum-weight vectors for the code over Fpof the design of points and hyperplanes of the affine geometry AGd(Fp), where p is a prime.

Maximal Weight Divisors of Projective Reed-Muller Codes

Projective Reed-Muller (PRM) codes, as the name suggests, are the projective analogues of generalized Reed-Muller codes. The parameters are known, and small steps have been taken towards pinning down the codeword weights that occur in any PRM code. We determine, for any PRM code, the greatest common divisor of its codeword weights.

Generalized Reed-Muller codes and power control in OFDM modulation

Controlling the peak-to-mean envelope power ratio (PMEPR) of orthogonal frequency-division multiplexed (OFDM) transmissions is a notoriously difficult problem, though one which is of vital importance for the practical application of OFDM in low-cost applications. The utility of Golay complementary sequences in solving this problem has been recognized for some time. In this paper, a powerful theory linking Golay complementary sets of polyphase sequences and Reed-Muller codes is developed. Our main result shows that any second-order coset of a q-ary generalization of the first order Reed-Muller code can be partitioned into Golay complementary sets whose size depends only on a single parameter that is easily computed from a graph associated with the coset. As a first consequence, recent results of Davis and Jedwab (see Electron. Lett., vol.33, p.267-8, 1997) on Golay pairs, as well as earlier constructions of Golay (1949, 1951, 1961), Budisin (1990) and Sivaswamy (1978) are shown to arise as special cases of a unified theory for Golay complementary sets. As a second consequence, the main result directly yields bounds on the PMEPRs of codes formed from selected cosets of the generalized first order Reed-Muller code. These codes enjoy efficient encoding, good error-correcting capability, and tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using small numbers of carriers.

Minimum-weight codewords as generators of generalized Reed-Muller codes

We establish the range of values of /spl rho/, where 0/spl les//spl rho//spl les/m(q-1), for which the generalized Reed-Muller code R/sub Fq/(/spl rho/, m) of length q/sup m/ over the field F/sub q/ of order q is spanned by its minimum-weight vectors.

A Gap in GRM Code Weight Distributions

The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over \gf{3}. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3^{m-2} in \grm_3(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+\Delta, where d is the minimum distance and \Delta is the largest integer dividing all weights occurring in the code.

Minimum Weight and Dimension Formulas for Some GeometricCodes

The geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed–Muller codes, but the geometric codes are, in general, not. We obtain values for the minimum weight of these codes in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory. Using Hamada‘s formula, we also show that the dimension of the dual of the code of a projective geometry design is a polynomial function in the dimension of the geometry.

Gray isometries for finite chain rings and a nonlinear ternary (36, 3/sup 12/, 15) code

Using tensor product constructions for the first-order generalized Reed-Muller codes, we extend the well-established concept of the Gray isometry between (Z/sub 4/, /spl delta//sub L/) and (Z/sub 2//sup 2/, /spl delta//sub H/) to the context of finite chain rings. Our approach covers previous results by Carlet (see ibid., vol.44, p.1543-7, 1998), Constantinescu (see Probl. Pered. Inform., vol.33, no.3, p.22-8, 1997 and Ph.D. dissertation, Tech. Univ. Munchen, Munchen, Germany, 1995), Nechaev et al. (see Proc. IEEE Int. Symp. Information Theory and its Applications, p.31-4, 1996) and overlaps with Heise et al. (see Proc. ACCT 6, Pskov, Russia, p.123-9, 1998) and Honold et al. (see Proc. ACCT 6, Pskov, Russia, p.135-41, 1998). Applying the Gray isometry on Z/sub 9/ we obtain a previously unknown nonlinear ternary (36, 3/sup 12/, 15) code.

Generalized Reed–Muller Codes and Curves with Many Points

Words of low weight in trace codes correspond to curves with many points and the same holds for subcodes of low weight via the fibre product construction. In 1996 Heijnen and Pellikaan gave an algorithm to determine a basis of minimum weight subcodes of generalized Reed–Muller codes. We show how this algorithm can be used to produce curves with many points and also some new families of curves which reach the Hasse–Weil upper bound.

Bases of Minimum-Weight Vectors for Codes from Designs

An explicit basis of incidence vectors for the p-ary code of the design of points and hyperplanes of the affine geometry AG m ( F p ) for any prime pand any integer m≥2 is obtained, which, as a corollary, gives a new elementary proof that this code is a generalized Reed–Muller code. In the proof a class of nonsingular matrices related to Vandermonde matrices is introduced.

Cyclic subcodes of generalized Reed-Muller codes

We consider certain subcodes of generalized Reed-Muller (GRM) codes, which we call homogeneous generalized Reed-Muller (HRM) codes. In general, they have a much better minimum distance than the GRM codes. The parameters of HRM codes are related to those of projective Reed-Muller (PRM) codes. Unlike most PRM codes, punctured HRM codes are cyclic. Under the trace map, HRM codes map to binary codes. These are in general much larger than classical RM codes, for the same minimum distance.

Generators and weights of polynomial codes

Berman and Charpin proved that all generalized Reed-Muller codes coincide with powers of the radical of a certain algebra. The ring-theoretic approach was developed by several authors including Landrock and Manz, and helped to improve parameters of the codes. It is important to know when the codes have a single generator. We consider a class of ideals in polynomial rings containing all generalized Reed-Muller codes, and give conditions necessary and sufficient for the ideal to have a single generator. The main result due to Glastad and Hopkins (Comment. Math. Univ. Carolin. 21, 371 – 377 (1980)) is an immediate corollary to our theorem.¶We also describe all finite quotient rings $ \Bbb Z/m\Bbb Z[x_1,\dots,x_n]/I $ which are commutative principal ideal rings where I is an ideal generated by univariate polynomials and then give formulas for the minimum Hamming weight of the radical and its powers in the algebra $\Bbb F[x_1,\ldots ,x_n]/(x_1^{a_1}(1-x_1^{b_1}), \ldots ,x_n^{a_n}(1-x_n^{b_n}))$ where $\Bbb F$ is an arbitrary field.

A New Generalization of Reed-Muller Codes

An error-correcting code can be defined as a set of functions mapping $P$, called the set of places, to $A$, called the alphabet. With classical Generalized Reed-Muller Codes, $P$ is an $m$-dimensional vector space $F^m$ over a finite field $F$, and $A$ is just the finite field $F$. Then, $C= \text{ GRM}(\nu,m)$ is defined to be the set of all functions from $P$ to $A$ which, when represented as a polynomial of minimal degree through Lagrange interpolation, (see [2], for example) has degree less than or equal to $\nu$. This procedure can be generalized. $C=A_{\nu}$ is taken to be an element of the filtration of some filtered $F$-algebra $B$. $A$ is another $F$-algebra, and $P = \text{HOM}_{ALG} (B,A)$. Then, $C=C_{\nu}(B,A)$ is the set of elements of $B_{\nu}$ viewed as functions from $P$ to $A$ via $b(x) := x(b)$ for $x\in P$ and $b\in B$.

Extendable Steiner designs from finite geometries

We describe a simple procedure for constructing Steiner 2-designs with the parameters of the designs of points and lines of a finite projective or affine geometry of dimension m⩾3; the codes of many of the designs constructed in this way will contain the code of the relevant finite-geometry design (a Reed-Muller or generalized Reed-Muller code). The designs can be extended to 3-designs provided that planes in the finite-geometry design extend.

On the Number of Points of Some Hypersurfaces in [formula omitted]nq

For generalized Reed–Muller codes, whenqis large enough, we give the second codeword weight, that is, the weight which is just above the minimal distance, and we also list all the codewords which reach this weight. To do this we have to study the number of points of some hypersurfaces and some arrangements of hyperplanes. We also present some properties of the Möbius function of these arrangements.

Ternary codes of steiner triple systems

The code over a finite field Fq of a design 𝒟 is the space spanned by the incidence vectors of the blocks. It is shown here that if 𝒟 is a Steiner triple system on v points, and if the integer d is such that 3d ≤ v < 3d+1, then the ternary code C of 𝒟 contains a subcode that can be shortened to the ternary generalized Reed-Muller code ℛF3(2(d − 1),d) of length 3d. If v = 3d and d ≥ 2, then C⟂ ⊆ ℛF3(1,d)⊆ ℛ F3(2(d − 1),d) ⊆ C. © 1994 John Wiley & Sons, Inc.

The automorphism group of Generalized Reed-Muller codes

We prove that the automorphism group of Generalized Reed-Muller codes is the general linear nonhomogeneous group. The Generalized Reed-Muller codes are introduced by Kasami, Lin and Peterson. An extensive study was made by Delsarte, Goethals and Mac-Williams; our result follows their description of the minimum weight codewords. An automorphism of a cyclic q-ary code is here a substitution over the field GF( q m ). In the more general case where the automorphisms are defined by monomial matrices, we also obtain the automorphism group (called the monomial group) as the direct product of the general linear nonhomogeneous group with the multiplicative group of the alphabet field.