Weight Codewords Research Articles

Codes with the same coset weight distributions as the Z/sub 4/-linear Goethals codes

We study the coset weight distributions of the family of Z/sub 4/-linear Goethals-like codes of length N=2/sup m+1/, m/spl ges/3 odd, constructed by Helleseth, Kumar, and Shanbhag (see Designs, Codes and Cryptography, vol.17, no.1-3, p.246-62, 1999). These codes have the same Lee weight distribution as the Z/sub 4/-linear Goethals code /spl Gscr//sub 1/, and, therefore (taking into account the result of Hammons, Kumar, Sloane, Calderbank, and Sole), the binary images of all these codes by the Gray map have the same weight distribution as the binary Goethals code. We prove that all these codes have the same coset weight distributions as the Z/sub 4/-linear Goethals code, constructed by Hammons, Kumar, Sloane, Calderbank, and Sole (see ibid., vol.40, p.301-19, March 1994). The cosets of weight four is the most difficult case. In order to find the number of codewords of weight four in a coset of weight four we have to solve a nonlinear system of equations over the Galois field GF(2/sup m/). Such a system (the degree of one of the equations) depends on k. We prove that the distribution of solutions to such a system does not depend on k and, therefore, coincides with the case k=1 considered earlier by Helleseth and Zinoviev (see Designs, Codes and Cryptography, vol.17, no.1-3, p.246-62, 1999). For k=1, we solved this system in the following sense: for all cases (of cosets of weight four) we have either an exact expression, or an expression in terms of the Kloosterman sums.

On the Non-Minimal codewords of weight 2dmin in the binary Reed-Muller Code

Abstract We compute the number of non-minimal codewords of weight 2 d min in the binary Reed-Muller code.

On the frame-error rate of concatenated turbo codes

Turbo codes with long frame lengths are usually constructed using a randomly chosen interleaver. Statistically, this guarantees excellent bit-error rate (BER) performance but also generates a certain number of low weight codewords, resulting in the appearance of an error floor in the BER curve. Several methods, including using an outer code, have been proposed to improve the error floor region of the BER curve. We study the effect of an outer BCH code on the frame-error rate (FER) of turbo codes. We show that additional coding gain is possible not only in the error floor region but also in the waterfall region. Also, the outer code improves the iterative APP decoder by providing a stopping criterion and alleviating convergence problems. With this method, we obtain codes whose performance is within 0.6 dB of the sphere packing bound at an FER of 10/sup -6/.

On binary cyclic codes with codewords of weight three and binary sequences with the trinomial property

Golomb and Gong (1997 and 1999) considered binary sequences with the trinomial property. In this correspondence we shall show that the sets of those sequences are (quite trivially) closely connected with binary cyclic codes with codewords of weight three. This approach gives us another way to deal with trinomial property problems. After disproving one conjecture formulated by Golomb and Gong, we exhibit an infinite class of sequences which do not have the trinomial property, corresponding to binary cyclic codes of length 2/sup m/-1 with minimum distance exactly four.

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.

New 3-designs from Goethals codes over [formula omitted

We consider t-designs constructed from codewords in the Goethals code G m over Z 4 . Some new designs are constructed from the support (size 7) of minimum Lee weight codewords for m=5 and m=7. The parameters of the designs are 3-(2 5,7,105) , 3-(2 5,7,7) and 3-(2 7,7,560) .

On covering radii and coset weight distributions of extremal binary self-dual codes of length 56

We present a method to determine the complete coset weight distributions of doubly even binary self-dual extremal [56, 28, 12] codes. The most important steps are (1) to describe the shape of the basis for the linear space of rigid Jacobi polynomials associated with such codes in each index i, (2) to describe the basis polynomials for the coset weight enumerators of the assigned coset weight i by means of rigid Jacobi polynomials of index i. The multiplicity of the cosets of weight i have a connection with the frequency of the rigid reference binary vectors v of weight i for the Jacobi polynomials. This information is sufficient to determine the complete coset weight distributions. Determination of the covering radius of the codes is an immediate consequence of this method. One important practical advantage of this method is that it is enough to get information on 8190 codewords of weight 12 (minimal-weight words) in each such code for computing every necessary information.

Elementary 2-group character codes

We describe a class of codes over GF(q), where q is a power of an odd prime. These codes are analogs of the binary Reed-Muller codes and share several features in common with them. We determine the minimum weight and properties of these codes. For a subclass of codes we find the weight distribution and prove that the minimum nonzero weight codewords give 1-designs.

New Optimal Self-Dual Codes over GF(7)

In this paper, double circulant self-dual codes over GF(7) are presented, including [12,6,6] codes which are new optimal codes. It is shown that the supports of the codewords of weights 9, 10 and 11 in double circulant [20,10,9] codes form 3-designs. For larger lengths, some good self-dual codes are constructed from weighing matrices.

Extremal double circulant Type II codes over Z 4 and construction of 5-(24, 10, 36) designs

In this paper, we provide a classification of extremal double circulant Type II codes of length 24 over Z 4. It is shown that the sets of supports of the codewords of Hamming weight 10 in certain extremal Type II codes of length 24 form 5-(24, 10, 36) designs. Designs with these parameters were not previously known to exist. A new extremal Type II double circulant code of length 32 is also constructed.

Construction of Extremal Type II Codes over \mbox{\zZ}_4

In this paper, we give a pseudo-random method to construct extremal Type II codes over\ZZ_4 . As an application, we give a number of new extremal Type II codes of lengths 24, 32 and 40, constructed from some extremal doubly-even self-dual binary codes. The extremal Type II codes of length 24 have the property that the supports of the codewords of Hamming weight 10 form 5-(24,10,36) designs. It is also shown that every extremal doubly-even self-dual binary code of length 32 can be considered as the residual code of an extremal Type II code over \ZZ_4.

New infinite families of 3-designs from preparata codes over Z 4

We consider t-designs constructed from codewords in the Preparata code P m over Z 4. A new approach is given to prove that the support (size 5) of minimum Lee weight codewords form a simple 3-design for any odd integer m ⩾ 3. We also show that the support of codewords with support size 6 form four new families of simple 3-designs, with parameters (2 m , 6, 2 m − 8), (2 m , 6, 5 · (2 m−1 − 4)), (2 m , 6,20 ·(2 m−1 − 4)/3) and (2 m , 6, 18 · (2 m−1 − 4)), for any odd integer m ⩾ 5. Codewords with support size 7 are also investigated by computer search.

Antichain Codes

We show that almost all codes satisfy an antichain condition. This states that the minimum length of a two dimensional subcode of a code C increases if the subcode is constrained to contain a minimum weight codeword. In particular, almost no code satisfies the chain condition. In passing, we study the typical behaviour of codes with respect to generalized distances and show that almost all lie on a generalized Varshamov-Gilbert bound.

Punctured turbo-codes for BPSK/QPSK channels

We discuss a procedure for designing high-rate turbo-codes via puncturing, with applications to BPSK/QPSK channels. Rates in the form k/(k+1), 2/spl les/k/spl les/16, are considered for constituent encoders with memory sizes m=3 and 4. The algorithm includes the selection of constituent encoder generator polynomials, puncture patterns, and interleavers with the goal of maximizing the minimum codeword weight for weight-two and weight-three inputs. The performance of the proposed codes are found via computer simulation and are observed in each case to be less than 0.9 dB (m=3), and 0.75 dB (m=4), from their respective channel capacity limits at a bit-error rate of 10/sup -5/. We consider two applications of punctured turbo-codes.

The Weight Enumerator of the Code of the Projective Plane of Order 5

We outline the determination of the weight enumerator of the code of the projective plane of order 5 by hand. The main tools used are a version of Gleason's theorem on the enumerators of self-dual codes and geometric descriptions of codewords of low weight. This paper is in part a condensation of the companion paper (Nota di Mat.) that contains full details.

An Infinite Family of 3-Designs from Preparata Codes over Z_4

We show that the support of minimum Lee weight codewords having Hamming weight 5 in the Preparata code over Z_4 form a 3-(2^m,5,10) design for any odd integer m \geq 3.

Codes of Small Defect

The parameters of a linear codeC over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1. Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d] -codes which have no code words of weight d+1 . Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.

On the binary codes of Steiner triple systems

The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple system STS(v) is studied. A sufficient condition for such a code to contain a unique equivalence class of STS(v)'s of maximal rank within the code is proved. The code of the classical Steiner triple system defined by the lines in PG(n-1,2) ( n\geq 3), or AG(n,3) ( n\geq 3) is shown to contain exactly v codewords of weight r = (v-1)/2, hence the system is characterized by its code. In addition, the code of the projective STS(2^{n}-1) is characterized as the unique (up to equivalence) binary linear code with the given parameters and weight distribution. In general, the number of STS(v)'s contained in the code depends on the geometry of the codewords of weight r. It is demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. This relation is utilized for the construction of some infinite classes of Steiner triple systems without ovals.

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.

Description of Minimum Weight Codewords of Cyclic Codes by Algebraic Systems

We consider cyclic codes of lengthnover Fq,nbeing prime toq. For such a cyclic codeC, we describe a system of algebraic equations, denoted by SC(w), wherewis a positive integer. The system is constructed from Newton's identities, which are satisfied by the elementary symmetric functions and the (generalized) power sum symmetric functions of the locators of codewords of weightw. The main result is that, in a certain sense, thealgebraicsolutions of SC(w) are in one-to-one correspondence with all the codewords ofChaving weight lower thanw. In the particular case wherewis the minimum distance ofC, all minimum weight codewords are described by SC(w). Because the system SC(w) is very large, with many indeterminates, no great insight can be directly obtained, and specific tools are required in order to manipulate the algebraic systems. For this purpose, the theory ofGröbner basescan be used. A Gröbner basis of SC(w) gives information about the minimum weight codewords.