Some Mysterious Sequences Associated with LDPC Codes

Abstract

One of the most important research areas in coding theory is weight enumeration. This is a large subject, but the basic problem is easily stated: determine or estimate the weight enumerator (B ₀, ..., B _n ) for an (n,k) binary linear code, specified by a (n − k) ×n parity-check matrix $\mathcal{H}$ with entries from GF(2). Here

$$B_i = \# \{c \in {GF(2)}^n: \mathcal{H}c^T = 0, {\mathop{\rm wt}}(c = i)\},$$

where ${\mathop{\rm wt}}(c)$ is the weight of the vector c. If the number of codewords is large, the logarithmic weight enumerator, i.e.,

$$ ({1 \over n}\log B_0, \ldots, {1 \over n}\log B_n)$$

is more convenient. If a code belongs to a family of codes which share similar properties, the log-weight enumerator may approach a limiting function called the spectral shape:

$${1 \over n} \log\left(B_{\lfloor\theta n\rfloor}\right)\rightarrow E(\theta), \qquad 0 < \theta <1.$$

In modern coding theory, $\mathcal{H}$ is usually very large and very sparse, e.g., the row and column sums are bounded as n → ∞. The corresponding codes are called low density parity-check codes. Often we are faced with large collections, or ensembles, of long LDPC codes, which share similar properties, in which case it may be difficult to find the spectral shape of an individual member of the ensemble, but relatively easy to calculate the ensemble average.

This research was supported by the Lee Center for Advanced Networking and the Sony Corporation.