A Tight Estimate for Decoding Error-Probability of LT Codes with Dense Rows

A novel approach of estimating Decoding Error-Probability (DEP) of LT codes, using the conditional Kovalenko’s Rank-Distribution, is proposed. The estimate is utilized for constructing optimal LT codes with dense rows achieving the DEP that is very close to Kovalenko’s FullRank Limit. Experimental evidences which show the viability of the estimate is also provided. INTRODUCTION AND BACKGROUNDS In Binary Erasure Channels (BEC) such as the Internet, the task of Luby Transform (LT) decoder is to recover the unique solution of a consistent linear system HX = β , β = (β1, . . . , βm) ∈ (F s 2) , (1) where H is an m × n matrix over F2. This can be explained briefly as follows. (For detailed backgrounds, consult with [1–3]). In LT codes, to communicate an information symbol vector α = (α1, . . . , αn) ∈ (F s 2) , a sender constantly generates and transmits a syndrome symbol βi = Hiα T over BEC, where Hi ∈ F2 is generated uniformly at random on the fly by using the Robust Soliton Distribution μ(x) = ∑ μdx d (see [1]). A receiver then acquires a set of pairs {(Hit , βit)} m t=1 and interprets it as system (1). In the system, the variable vector X = (x1, . . . , xn) ∈ (F s 2) n represents the information symbol vector α, the column-dimension n of H is fixed, and the row-dimension m is a variable. Thus, a reception overhead γ = m−n n is the key parameter for measuring error-performance of LT codes. System (1) has its unique solution, iff, Rank(H) = n the full rank of H . In case of the fullrank, the unique solution can be recovered by using a Maximum-Likelihood Decoding Algorithm (MLDA) such as the ones in [3,6]. These algorithms are an efficient Gaussian Elimination (GE) that fully utilize an approximate lower triangulation of H , which is obtainable by using the diagonal extension process with various greedy algorithms [2, 3]. Under those GE based MLDAs, thus, the probability of decoding success is the full-rank probability Pr(Rank(H) = n). Let us define the Decoding Error Probability (DEP) of an LT code generated by a row-degree distribution ρ(x) = ∑ ρdx d as the rank-deficient probability Pr err(1 + γ, n, ρ) = 1 − Pr(Rank(H) = n), (2) where H is an m× n check matrix of system (1) generated by ρ(x) with m = (1 + γ)n. Then for a given error-bound (or deficiency bound) 0 ≤ δ ≤ 1, define γmin(δ, n, ρ) = min γ≥0 {γ | Pr err(1 + γ, n, ρ) < δ}, (3) and refer to as the Minimum Stable Overhead (MSO) of a code within the error-bound δ. By their definitions, obviously, Pr err(1 + γ, n, ρ) ≤ δ for any γ ≥ γmin(δ, n, ρ). It was shown in [8] by the authors of this paper that probabilistic lower-bounds for DEP and MSO exist, refered to as Kovalenko’s Full-Rank Limit and its Overhead (KFRL and KFRO hereafter). Specifically, KFRL is the function K(1 + γ, n) = 1 − n ∏ i=k+1 ( 1 − 1 2i ) , k = γn, (4) and is less than Pr err(1 + γ, n, ρ). Similar to MSO, KFRO is the minimum γ defined as γK(δ, n) = min γ {γ ≥ 0 |K(1 + γ, n) ≤ δ}. (5) Designers may tell how closely the DEP can approach the ideal limit KFRL by comparing MSO and KFRO. Without loss of generality, thus, the key objective of designing good codes is to obtain a ρ(x), by which DEP and MSO of generated codes are close to KFRL and KFRO, respectively, while a random H in system (1) is as sparse as possible for decoding efficiency. It was also shown in [8] experimentally that, even for short n and small γ, LT codes may achieve the DEP and MSO that are close to KFRL and KFRO, respectively, when a small fraction of rows of degree n 2 is supplemented to μ(x). The main obstacle in designing codes with dense rows was however that an appropriate value of the fraction was empirically known by exhaustive experiments only. Furthermore, behaviors of the codes such as error-floor regions in DEP were not predictable in advance, unless their DEP is explicitly traced out through experiments. (It may require few weeks to complete the experiments on ordinary computers). In the next section, what shall be discussed is a novel approach of estimating DEP of LT codes via the conditional Kovalenko’s rank-distribution of binary random matrices. The approach is very practical in that, for any assigned value of the fraction, the estimate is very close to the DEP, and also, it can be computed explicitly right away (say within a second). For any destined error-bound δ, thus, the optimal dense fraction can be found for the constraint Pr err(1+γ, n, ρ) ≤ δ without exhaustive experiments. Experimental evidences which show the viability of the estimate are also provided. RECENT ADVANCES: THE ESTIMATE FOR DEP OF LT CODES For a given μ(x) = ∑ d≤d0 μdx , limn→∞ d0 n = 0, let ρ(x) be a supplementation of μ(x) with κ ≥ 0 in such a way that