LT Codes from an Arranged Encoder Matrix and Degree Distribution Design with Dense Rows

In this paper, we present the design of LT degreedistributions ρ(x) supplemented with dense fractions by which the generated LT codes [1] of short block lengthsn may fit for the Separated Maximum-Likelihood Decoding Algorithm (SMLDA) demonstrated in [7], [8]. By using the Kovalenko’s Rank Distribution [12]–[14], we also present the rank distribution of random matrices generated by the designedρ(x). For LT encoding, we use a(kn) × n encoder matrix M over F2 whose row-degree distribution follows a designedρ(x). We then substantiate, based on our extensive experimental results, th at a stable overheadγ of codes for the successful S-MLDA can be significantly improved when dense rows are supplemented to M , while the S-MLDA maintains its complexity of decoding in symbol additions within few tens ofn. I. I NTRODUCTION AND BACKGROUNDS Let α = (α1, . . . , αn) ∈ (F2) be a given input symbol vector that we would like to communicate to receivers over th Binary Erasure Channels (BEC). In the original LT transmission scheme in [1], an encoder constantly generates encodin g symbolsβi ∈ F2 with α and the Robust Soliton Distribution (RSD) μ(x) = ∑ μdx d in such a way that 1) a degreed is randomly chosen with probability μd; 2) a rowHi ∈ F2 of degreed is randomly chosen from ( n d ) possible choices; 3) thenβi = Hi · α is transmitted over the BEC. The transmission stops when a receiver acquires enough number of encoding symbols, say the acquired symbols as β = (β1, . . . , βm). At a receiver end, an LT decoder recovers the α by solving a consistent linear system HX = β , β = (β1, . . . , βm) ∈ (F2) (I.1) wherem = (1 + γ)n for someγ > 0 and H is an m × n matrix that consists of rows Hi satisfyingHiα = βi. It is shown that, by Luby [1,Theorem 17], if γ ≥ ln(n/δ)/ √ n, then system (I.1) can be solved for α uniquely by using the Message Passing Algorithm (MPA) in [10] with probability at least1− δ. For short block lengthsn, however, a largerγ is required for the successful MPA. WhenH is designed by the RSD with short block lengths n within several hundreds, we observed that the rank deficienc y η = Ker(H) is feasibly happened and is less than a few. A small deficiencyη can be removed by using the pre-coding strategies as in Raptor codes [4] but with a small degradatio n in overheadsγ. In contrast, supplementing dense fractions in μ(x) can increasePr(Rank(H) = n) rapidly without extra costs inγ. In our earlier work in [8], we designed LT codes with dense rows that fit for the block lengths n, 10 ≤ n ≤ 10, with the S-MDLA. Particularly, a very small dense fractionρn/2 = 0.005 was supplemented to μ(x), and a stableγ for the successful S-MLDA was less than 0.015. With shorter block lengthsn in several hundreds, however, it seems to us that much larger ρn/2 is required for a stable γ. In section V, we design a distribution ρ(x) with ρn/2 = 0.082. Dense rows may cause a nontrivial drawback in communicating H to a decoder. Note that, for every instance of α, a decoder should identify H first by using the same random generator of an encoder. Or, each Hi should be delivered to a receiver withβ. In both cases, due to dense rows, the cost in communicatingH may not be trivial. Supplemented dense rows also may degradate the computational efficiency of the MLDA in [3] seriously. Nonetheless, H can be approximate lower triangulated by using the Approximate Lower Triangulation Algorithm (ALTA) in [2], [3] even with theγ very close to 0. Furthermore, most of the dense rows become redundant. Therefore, those redundant rows should be identified so that symbol additions over the redundant rows can be removed by using the S-MLDA [7], [8]. In this paper, we design LT codes for short block length n in two perspectives. First, we alter a designed RSD μ(x) into a ρ(x) = ∑ ρdx d by supplementing a dense fraction ρn/2 for the higherPr(Rank(H) = n). Thus, even for short block lengthsn andγ close to0, a check matrixH of system (I.1) may have its full column rank n with high probability. Second, to communicateH to a decoder efficiently, we use a (kn)×n arranged matrixM for LT encoding, so that at a receiver end, an LT decoder can identify H simply by reading rows of M . With the arranged encoder matrix M , designed with an altered row-degree distribution ρ(x), a decoder can quickly identify H by reading rows ofM without extra cost in communicatingH to a decoder. In this encoding scheme, however, a row-degree distributions of H may be deviated from ρ(x) the distribution of the encoder M seriously. Thus, the random features of the original encoding scheme should be limited. In particular, a stable overhead γ of codes for the successful MPA may be degraded seriously. Nonetheless, our simulation exhibits that the degradation does not affect th e performance inγ under the S-MLDA. (Compare black and red curves inFig. V.1 in section V). Due to the fixedM , the fixed block-lengthn could be also drawback to the flexibility of block lengthsn. However, if changes inn are not so severe, then this constraint can be negotiated by using shortening techniques with null symbol s on α and resizing the symbol-size s. The rest of the paper is organized as follows. In section II, we introduce the S-MLDA of [7], [8] in a comprehensive level of linear algebra. In section III, we estimate the complexit y of the S-MLDA by counting the number of {sign, bit}-flips and symbol additions. In section IV, we design an LT degree distributionρ(x) by supplementing dense fractions on μ(x) for higherPr(Rank(H) = n). Also, in this section, we introduce Kovalenko’s rank distribution of random matrices over Fq in [12]–[14]. In section V, we present our experimental result s tested with LT codes generated by ρ(x) for short block lengths n, 10 ≤ n ≤ 10, for the following scenarios: 1) the decoding error rate of codes under the S-MLDA and the MPA, which check matrices H are generated from a 5n × n arranged encoder matrix M and generated by a pseudo-random generator; 2) the number of symbol additions of β by the S-MLDA and the original MLDA in [3]; 3) fraction of references n to tell the efficiency of the Gaussian Elimination (GE) of the S-MLDA; 4) the rank-deficiencyη = dim(Ker(H)) when the SMLDA fails to recover the input symbol vector α. We then summarize the paper in section VI. II. T HE SEPARATED MLDA Let us first clarify several notations that are used in the remainder of the paper. For a given m × n matrix K over F2, we denotekij , Ki, K , and |K| as the(i, j) entry, the i row, the k column, and the number of 1’s of K, respectively. We use the notation 1ij to indicate the(i, j) 1 in K. With the matrix dimensionm×n, let [m] = {1, 2, . ,m} and [n] = {1, 2 . , n} the row and column index set of K, respectively. Let us first consider a conventional GE [15], [16] with system (I.1). The GE onH is an LU -factorization process that returnsPHQ = LU such that:L andU is in an lower and upper triangular form, respectively; and (P,Q) is a pair of row and column permutation matrices of H. In fact, both L andU are obtained in a product form such that