Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

Explicit Computation of Generalized Hamming Weights for Some Algebraic Geometric Codes

The a-invariant, the defining ideal, the dimension and the minimal distance of some Reed-Muller type codes arising from the Veronese variety over a finite field are determined. Some examples are provided to illustrate the main results. These codes are a natural generalization of the projective Reed-Muller codes.

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

Firstly, we give a formula on the generalized Hamming weight of linear codes constructed generically by defining sets. Secondly, by choosing properly the defining set we obtain a class of cyclotomic linear codes and then present two alternative formulas to calculate their generalized Hamming weights. Lastly, we determine their weight distribution and generalized Hamming weights partially. Especially, we solved the generalized Hamming weights completely in one case.

The peak-to-mean envelope power ratio (PMEPR) of a code employed in\northogonal frequency-division multiplexing (OFDM) systems can be reduced by\npermuting its coordinates and by rotating each coordinate by a fixed phase\nshift. Motivated by some previous designs of phase shifts using suboptimal\nmethods, the following question is considered in this paper. For a given binary\ncode, how much PMEPR reduction can be achieved when the phase shifts are taken\nfrom a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the\nachievable PMEPR is established, which is related to the covering radius of the\nbinary code. Generally speaking, the achievable region of the PMEPR shrinks as\nthe covering radius of the binary code decreases. The bound is then applied to\nsome well understood codes, including nonredundant BPSK signaling, BCH codes\nand their duals, Reed-Muller codes, and convolutional codes. It is demonstrated\nthat most (presumably not optimal) phase-shift designs from the literature\nattain or approach our bound.\n

The generalized hamming weight played an important role in coding theory. In the study of the wiretap channel of type II, the generalized hamming weight was extended to a two-code format. Two equivalent concepts of the generalized hamming weight hierarchy and its two-code format, are the inverse dimension/length profile (IDLP) and the inverse relative dimension/length profile (IRDLP), respectively. In this paper, the relations between the IDLP and the IRDLP are investigated. By using a quotient subcode set, the singleton bound on the IRDLP is improved by different methods. The new bounds are useful for the study of a coordinated two-party wiretap channel of type II.

Comparing with hard decision decoding algorithms, soft decoding has a lower probability of bit error but a higher computational complexity. As a maximum-likelihood soft decoding method, the <i>A</i>* algorithm is the most basic and widely used to minimize bit error probability. However, its average computational complexity strongly depends on a seed codeword and a heuristic function utilized during the decoding process. To efficiently reduce the computational complexity while maintaining the decoding accuracy theoretically and practically, this paper proposes an improved <i>A</i>* decoding algorithm consisting of two phases. The first phase applies the greedy list decoding to the linear block code to obtain a seed codeword. According to the seed, the second phase applies the improved <i>A</i>* algorithm to obtain the final decoding output. The heuristic function used in the <i>A</i>* algorithm is modified in two aspects: (1) use more information of partial decoded symbols to improve the accuracy of the function and (2) take advantage of Hamming distance to reduce the search space. Simulations on the <i>RM</i>(5, 2) Reed-Muller codes and [128, 64] binary extended BCH code show that this improved <i>A</i>* algorithm is more efficient in average decoding complexity than many other algorithms while maintaining the decoding accuracy.

Let \(\mathcal R_{t,n}\) denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary Reed-Muller code RM(2,6) in the sets \(\mathcal R_{t,6}\), t=0,1,2 is 16. Second, we show that the covering radius of the binary Reed-Muller code RM(2,7) in the set \(\mathcal R_{3,7}\) is 32. We derive a new lower bound for the covering radius of the Reed-Muller code RM(2,n) in the set \(\mathcal R_{n-1,4}\). Finally, we present new lower bounds in the sets \(\mathcal R_{t,7}\), t=0,1,2.

The algebraic analysis of linear code parameters reveals deep connections with cryptographic constructions, including the information dispersal algorithms (IDAs) and secret-sharing schemes. In this work, we propose an algebraic method for constructing bases of binary linear codes from subsets of codewords selected according to their generalized Hamming weights (GHWs). The approach employs a degree-compatible monomial ordering on the polynomial ring F2[x1, . . . , xn] and imposes the conditions d1(C) = 1 and dk (C) = n. Under these assumptions, we prove the existence of a generator matrix containing an invertible k × k submatrix, which guarantees correct information reconstruction. This structural property enables the direct application of binary linear codes to information dispersal and recovery mechanisms without the need for larger finite fields. We validate the proposed framework through algebraic proofs and an explicit example illustrating both the dispersal and recovery procedures. These results provide a theoretical foundation for the design of information dispersal schemes relying exclusively on binary linear codes.

A new class of binary codes is introduced. The codes have length n = 1.5 · 2m (m ≥ 3), and have a similar ‘double inductive’ structure to that possessed by the class of Reed-Muller (RM) codes. The new codes are structured such that their length-to-distance ratio n/d = 2m, m ≥ 2. For rates less than 1/2, the new codes have a higher rate than the next Reed-Muller code of the same order and length n = 2 · 2m.

We design 4-ary codebooks using Reed-Muller codes for multiple-input-multiple-output (MIMO) beamforming systems. An upperbound on the maximum correlation magnitude metric of a 4-ary codebook is derived from the minimum Hamming distance of the corresponding binary linear block code. The 4-ary MIMO beamforming codebooks for the number of transmit antennas equal to 4, 8, 16, and 32 are tabulated. We show that the 4-ary codebook using Reed-Muller codes provides the same symbol error probability (SEP) as the Long-Term Evolution (LTE) codebook when the number of transmit antennas is 4 and when the number of feedback bits is 4.

Restrictions on weight distribution of Reed-Muller codes

A group code is an ideal in a suitable group algebra. Cyclic codes, Reed-Muller codes, the binary extended Golay code and many other remarkable codes are group codes. In this paper we answer the following question: Under what conditions do exist self-dual group codes over Galois rings?.

Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently, due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be testable with O(N/d) queries, and has a dimension of a N − (log N)log d. The polylogarithmically small co-dimension is the basis of constructions of small set expanders with many bad eigenvalues, and size-efficient PCPs based on a shorter version of the long code. The smallest possible co-dimension for a distance d code (without any testability requirement) is a d/2 log N, achieved by BCH codes. This raises the natural question of understanding where in the spectrum between the two classical families, Reed-Muller and BCH, the optimal co-dimension of a distance d LTC lies --- in other words the price one has to pay for local testability.One promising approach for constructing LTCs is to focus on affine-invariant codes, whose structure makes testing guarantees easier to deduce than for general codes. Along these lines, the authors of [HRZS13] and [GKS13] recently constructed an affine-invariant family of high-rate LTCs with slightly smaller co-dimension than Reed-Muller codes. In this work, we show that their construction is essentially optimal among linear affine-invariant LTCs that contain the Reed-Muller code of the appropriate degree.