Reed-Muller Code RM Research Articles

The Covering Radius of the Third-Order Reed-Muller Code RM(3,7) is 20

We prove the covering radius of the third-order Reed-Muller code RM(3, 7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of RM(6, 6)/RM(3, 6), we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove that any function in the remaining 4 types can be transformed into a type (6,10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve the third-order nonlinearity 21 (with the assistance of computers). By the way, we prove that the affine transformation group over any finite field can be generated by two elements.

Classification of the Codewords of Weights 16 and 18 of the Reed-Muller Code RM(n-3,n)

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary rth-order Reed-Muller code RM(r, n) can be viewed as the set of all n-variable Boolean functions of algebraic degree at most r. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remain the best ones in the literature. The Reed-Muller code RM(n -3, n), which corresponds to the dual of the Reed-Muller code RM(2, n), has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance 2n-r due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of RM(n -3, n) whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code RM(n -3, n). We provide a classification of the codewords of weight 16 and 18 of the Reed-Muller code RM(n -3, n). Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly.

New bounds on the covering radius of the second order Reed-Muller code of length 128

In 1981, Schatz proved that the covering radius of the binary Reed-Muller code RM(2, 6) is 18. It was previously shown that the covering radius of RM(2, 7) is between 40 and 44. In this paper, we prove that the covering radius of RM(2, 7) is at most 42. As a corollary, we also find new upper bounds for RM(2, n), n = 8, 9, 10. Moreover, we give a sufficient and necessary condition for the covering radius of RM(2, 7) to be equal to 42. Using this condition, we prove that the covering radius of RM(2, 7) in RM(4, 7) is exactly 40, and as a by-product, we conclude that the covering radius of RM(2, 7) in the set of 2-resilient Boolean functions is at most 40, which improves the bound given by Borissov et al. (IEEE Trans. Inf. Theory 51(3):1182–1189, 2005).

On the covering radius of the third order Reed–Muller code RM(3, 7)

The covering radius of the third order Reed–Muller code of length 128 has been an open problem for many years. The best upper bound of it is known to be 22. In this paper, we give a sufficient and necessary condition for the covering radius of RM(3, 7) to be equal to 22. Using this condition, we prove that the covering radius of RM(3, 7) in RM(4, 7) is 20. Therefore, if the third-order nonlinearity of a 7-variable Boolean function is greater than 20, then its algebraic degree is at least 5. As a corollary, we conclude that the covering radius of RM(3, 7) in the set of 2-resilient Boolean functions is at most 20 which improves the bound given by Borissov et al. (IEEE Trans Inf Theory 51:1182–1189, 2005).

Spherically Punctured Reed-Muller Codes

Consider a binary Reed–Muller code RM $(r,m)$ defined on the $m$ -dimensional hypercube $\mathbb {F}_{2}^{m}$ . In this paper, we study punctured Reed–Muller codes $P_{r}(m,b)$ , whose positions are restricted to the $m$ -tuples of a given Hamming weight $b$ . In combinatorial terms, this paper concerns $m$ -variate Boolean polynomials of any degree $r$ , which are evaluated on a Hamming sphere of some radius $b$ in $\mathbb {F}_{2}^{m}$ . Codes $P_{r}(m,b)$ inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical $b$ -layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes $P_{r}(m,b)$ and find their distances for the arbitrary values of parameters $r,m$ , and $b$ . Finally, we describe recursive (successive cancellation) decoding of these codes.

Construction of balanced even-variable Boolean functions with optimal algebraic immunity

Given a positive integer n, let and . The generator matrix of the kth-order Reed–Muller code RM is an important tool in the study of Boolean functions' algebraic immunity. In this paper, choosing the last s column vectors in as a basis of the vector space , we study the values of the coefficients in the linear expressions of 's column vectors over this basis. As an application, we present a new construction of balanced even-variable Boolean functions on with optimal algebraic immunity by modifying the outputs of majority function. The nonlinearities of these constructed functions are also determined.

On the Optimum Cyclic Subcode Chains of <inline-formula> <tex-math notation="TeX">${\cal RM}(2, m)^{\ast}$</tex-math></inline-formula> for Increasing Message Length

The distance profiles of linear block codes can be employed to design variational coding scheme for encoding message with variational length and getting lower decoding error probability by large minimum Hamming distance, where one example is in the design of transport format combination indicators (TFCIs) in CDMA. Considering convenience for encoding, we focus on the distance profiles with respect to cyclic subcode chains (DPCs) of cyclic codes over GF(q) with length n such that gcd(n, q) = 1. In this paper, the optimum DPCs and the corresponding optimum cyclic subcode chains are investigated on the punctured second-order Reed-Muller code RM(2, m)* for increasing message length, where two standards on the optimums are studied according to the rhythm of increase. Ignoring the dimension profile, the device will coincide with that of TFCI.

Spherically Punctured Biorthogonal Codes

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.

On the Cusick–Cheon Conjecture About Balanced Boolean Functions in the Cosets of the Binary Reed–Muller Code

In this paper, an amplification of the Cusick-Cheon conjecture on balanced Boolean functions in the cosets of the binary Reed-Muller code RM(k,m) of order k and length 2m, in the cases where k = 1 or k ges (m-1)/2, is proved.

List Decoding of Biorthogonal Codes and the Hadamard Transform With Linear Complexity

Let a biorthogonal Reed-Muller code RM (1,m) of length n = 2m be used on a memoryless channel with an input alphabet plusmn1 and a real-valued output R. Given any nonzero received vector y in the Euclidean space Rn and some parameter epsiisin(0,1), our goal is to perform list decoding of the code RM (1, m) and retrieve all codewords located within the angle arccos e from y. For an arbitrarily small epsi, we design an algorithm that outputs this list of codewords with the linear complexity order of n [ln2 isin] bit operations. Without loss of generality, let vector y be also scaled to the Euclidean length radic(n) of the transmitted vectors. Then an equivalent task is to retrieve all coefficients of the Hadamard transform of vector y whose absolute values exceed nisin. Thus, this decoding algorithm retrieves all ne-significant coefficients of the Hadamard transform with the linear complexity n [ln2 isin] instead of the complexity n In2n of the full Hadamard transform.

On the second weight of generalized Reed-Muller codes

Not much is known about the weight distribution of the generalized Reed-Muller code RM q (s,m) when q > 2, s > 2 and m ? 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m --- 1)(q --- 1) we then find the first s + 1 --- (m --- 1)(q---1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Grobner basis theoretical methods.

On ℤ4-linear codes with the parameters of Reed-Muller codes

For any pair of integers r and m, 0 ? r ? m, we construct a class of quaternary linear codes whose binary images under the Gray map are codes with the parameters of the classical rth-order Reed-Muller code RM(r, m).

On the Covering Radii of Binary Reed–Muller Codes in the Set of Resilient Boolean Functions

Let R/sub t,n/ be the set of t-resilient Boolean functions in n variables, and let /spl rho//spl circ/(t,r,n) be the maximum distance between t-resilient functions and the rth-order Reed-Muller code RM(r,n). We prove that /spl rho//spl circ/(t,2,6)=16 for t=0,1,2 and /spl rho//spl circ/(3,2,7)=32, from which we derive the lower bound /spl rho//spl circ/(t,2,n) /spl ges/ 2/sup n-2/ with t /spl les/ n-4. Using a result from coding theory on the covering radius of (n-3)th- and (n-4)th-order Reed-Muller codes, we establish exact values of the covering radius of RM(n-3,n) in the set of 1-resilient Boolean functions in n variables, when /spl lfloor/n/2/spl rfloor/=1 mod 2 and lower bounds of RM(n-4,n) in the set of 2-resilient Boolean functions in n variables. This result leads again to different lower bounds for general dimensions n and r=0 or 3 mod 4.

New Covering Radius of Reed–Muller Codes for&amp;lt;tex&amp;gt;$t$&amp;lt;/tex&amp;gt;-Resilient Functions

In this paper, we introduce a new covering radius of RM(r,n) from cryptography viewpoint. It is defined as the maximum distance between t-resilient functions and the rth order Reed-Muller code RM(r,n). We next derive its lower and upper bounds. We further present a table of numerical data of our bounds.

Linear tail-biting trellises, the square-root bound, and applications for Reed-Muller codes

Linear tail-biting trellises for block codes are considered. By introducing the notions of subtrellis, merging interval, and sub-tail-biting trellis, some structural properties of linear tail-biting trellises are proved. It is shown that a linear tail-biting trellis always has a certain simple structure, the parallel-merged-cosets structure. A necessary condition required from a linear code in order to have a linear tail-biting trellis representation that achieves the square root bound is presented. Finally, the above condition is used to show that for r/spl ges/2 and m/spl ges/4r-1 or r/spl ges/4 and r+3/spl les/m/spl les/[(4r+5)/3] the Reed-Muller code RM(r, m) under any bit order cannot be represented by a linear tail-biting trellis whose state complexity is half of that of the minimal (conventional) trellis for the code under the standard bit order.

On the depth distribution of linear codes

The depth distribution of a linear code was recently introduced by T. Etzion (see ibid., vol.43, pp.1361-3, July 1997). In this correspondence, a number of basic and interesting properties for the depth of finite words and the depth distribution of linear codes are obtained. In addition, we study the enumeration problem of counting the number of linear subcodes with the prescribed depth constraints, and derive some explicit and interesting enumeration formulas. Furthermore, we determine the depth distribution of Reed-Muller code RM (m,r). Finally, we show that there are exactly nine depth-equivalence classes for the ternary [11,6,5] Golay codes.

The covering radius of Hadamard codes in odd graphs

The use of odd graphs has been proposed as fault-tolerant interconnection networks. The following problem originated in their design: what is the graphical covering radius of an Hadamard code of length 2 k − 1 and size 2 k − 1 in the odd graph O k ? Of particular interest is the case of k=2 m −1, where we can choose this Hadamard code to be a subcode of the punctured first order Reed-Muller code RM(1, m). We define the w-covering radius of a binary code as the largest Hamming distance from a binary word of Hamming weight w to the code. The above problem amounts to finding the k-covering radius of a (2 k,4 k, k−1) Hadamard code. We find upper and lower bounds on this integer, and determine it for small values of k. Our study suggests a new isomorphism test for Hadamard designs.