On the weight structure of Reed-Muller codes

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.

On the weight enumeration of weights less than 2.5 d of Reed—Muller codes

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring Z4 is developed. Our approach is based on the classical Brouwer-Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the Z4-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all Z4-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the Z4-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the Z4-linear XQR-code of length 80 is a new example of an extremal Z4-linear typeII code. Additionally, we give the symmetrized weight enumerator of the Z4-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the Z4-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze et al.

The idea to use error-correcting codes in order to construct public key cryptosystems was published in 1978 by McEliece [ME1978]. In his original construction, McEliece used Goppa codes, but various later publications suggested the use of different families of error-correcting codes. The choice of the code has a crucial impact on the security of this type of cryptosystem. Some codes have a structure that can be recovered in polynomial time, thus breaking the cryptosystem completely, while other codes have resisted attempts to cryptanalyze them for a very long time now. In this thesis, we examine different derivatives of the McEliece cryptosystem and study their structural weaknesses. The main results are the following: In chapter 3 we devise an effective structural attack against the McEliece cryptosystem based on algebraic geometry codes defined over elliptic curves. This attack is inspired by an algorithm due to Sidelnikov and Shestakov [SS1992] which solves the corresponding problem for Reed-Solomon codes. The presented algorithm is heuristic polynomial time and thus inverts trapdoors even for very large codes. In chapter 4, we show that the Sidelnikov cryptosystem [S1994], which is based on binary Reed-Muller codes, is insecure. The basic idea of our attack is to use the fact that minimum weight words in a Reed-Muller code have very particular properties. This attack relies on the ability to find minimum weight words in the code, a problem that is, in this specific instance, much easier than general decoding, and feasible for interesting parameters in a modest amount of time. The attack has subexponential running time if the order of the code is kept fixed, and it breaks the large keys as proposed by Sidelnikov in under an hour on a stock PC. In the chapter 5, we finally discuss some of the problems to solve if one attempts to generalize these algorithms.

Some construction techniques for self-dual codes are investigated, and the authors construct a singly-even self-dual (48,24,10)-code with a weight enumerator that was not known to be attainable. It is shown that there exists a singly-even self-dual code C' of length n=48 and minimum weight d=10 whose weight enumerator is prescribed in the work of J.H. Conway et al. (see ibid., vol.36, no.5, p.1319-33, 1990). Two self-dual codes of length n are called neighbors, provided their intersection is a code of dimension (n/2)-1. The code C' is a neighbor of the extended quadratic residue code of length 48.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.

The minimum pseudocodeword weight <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> of a linear graph-based code is more influential in determining decoding performance when decoded via iterative and linear programming decoding algorithms than the classical minimum distance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> under standard maximum-likelihood decoding Moreover, unlike the minimum distance which is unique to the code regardless of representation, the set of pseudocodewords, and therefore also the minimum pseudocodeword weight, depends on the graph representation used in decoding as well as on the communication channel. This means that a judicious choice of parity-check matrix is crucial for realizing the best potential of any graph-based code. In this paper, we introduce the notion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pseudoweight</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">redundancy</i> for the memoryless binary symmetric channel (BSC). Analogous to the stopping redundancy in the literature, this parameter gives the smallest number of rows needed for a parity-check matrix to have <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> . We provide some upper bounds on the BSC-pseudoweight redundancy and illustrate the concept with some results for Hamming codes, tree-based and finite geometry LDPC codes, Reed-Muller codes and Hadamard codes.

A generalized definition of higher weights for codes over finite chain rings and principal ideal rings and bounds on the minimum higher weights in this setting are given. Using this we generalize the definition for higher MDS and MDR codes. Computationally, the higher weight enumerator of lifted Hamming and Simplex codes over $${\mathbb{Z}_4}$$, the minimum higher weights for the lifted code of the binary [8,4,4] self-dual extended Hamming code, the lifted code of the ternary [12,6,6] self-dual Golay code and the lifted code of the binary [24,12,8] self-dual Golay code are given. Joint weight enumerators are used to produce MacWilliams relations for specific higher weight enumerators.

On the enumeration of self-dual codes

We consider the binary images of (8, 5) shortened cyclic codes. The (8, 5) shortened cyclic codes have a variety of choices. We have generated about 30000 sample codes with different weight distributions. Let S/sub w/ denote the set of generated sample codes with minimum weight w. The largest minimum weight of sample codes is 8. Let A/sub w/ denote the number of codewords of weight w of a sample code. In S/sub 7/, the smallest of A/sub 7/ is 10 for 10 sample codes and the second smallest of A/sub 7/ is 11 for six sample codes. In S/sub 7/, the smallest of A/sub 8/ is 728 for a sample code and the second smallest of A/sub 8/ is 729 for a sample code. We have chosen two sample codes from each of S/sub 7/ and S/sub 8/ which have the smallest and the second smallest sums of A/sub w/ for 7/spl les/w/spl les/9 in S/sub 7/ and S/sub 8/, respectively. For the AWGN channel using BPSK signaling, we have made simulation to evaluate the decoding error probabilities by a soft-decision decoding based on ordered statistics for the chosen four sample codes at SNR 2.0 to 5.0. These error probabilities are considerably smaller than the optimum error probabilities for (64, 40) subcodes of (64, 42) Reed-Muller code.

First it is shown that all binary Reed-Muller codes with one digit dropped can be made cyclic by rearranging the digits. Then a natural generalization to the nonbinary case is presented, which also includes the Reed-Muller codes and Reed-Solomon codes as special cases. The generator polynomial is characterized and the minimum weight is established. Finally, some results on weight distribution are given.

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{n}</tex> denote the number of cosets with minimum weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2^{m}, m + 1)</tex> Reed-Muller code. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{n}</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{m-2} \leq n &lt; 2^{m-2} + 2^{m - 4}</tex> is determined.

Certain \(\mathbb{Z}S_n\)-modules related to the kernels ofincidence maps between types in the poset defined by the natural productorder on the set of n-tuples with entries from {1, \(\ldots\) ,m} are studied as linear codes (whencoefficients are extended to an arbitrary field K). Theirdimensions and minimal weights are computed. The Specht modules areextremal among these submodules. The minimum weight codewords of theSpecht module are shown to be scalar multiples of polytabloids. Ageneralization of t-design arising from the natural permutationS n-modules labelled by partitions with mparts is introduced. A connection with Reed-Muller codes is noted and acharacteristic free formulation is presented.

Let R(r, m) be the rth order Reed-Muller code of length 2m, and let ρ(r, m) be its covering radius. We prove that if 2≤ k ≤ m - r - 1, then ρ(r + k, m + k) ≥ ρ(r, m + 2(k - 1). We also prove that if m - r ≥ 4, 2 ≤ k ≤ m - r - 1, and R(r, m) has a coset with minimal weight ρ(r, m) which does not contain any vector of weight ρ(r, m) + 2, then ρ(r + k, m + k) ≥ ρ(r, m) + 2k(. These inequalities improve repeated use of the known result ρ(r + 1, m + 1) ≥ ρ(r, m).

In this paper, we give some new extremal ternary self-dual codes which are constructed by skew-Hadamard matrices. This has been achieved with the aid of a recently presented modification of a known construction method. In addition, we survey the known results for self-dual codes over GF (5) constructed via combinatorial designs, i.e. Hadamard and skew-Hadamard matrices, and we give a new self-dual code of length 72 and dimension 36 whose minimum weight is 16 over GF (5) for the first time. Furthermore, we give some properties of the generated self-dual codes interpreted in terms of algebraic coding theory, such as the orders of their automorphism groups and the corresponding weight enumerators.