Linear-programming Decoding of Non-binary Linear Codes

Of course, within such a framework it is desirable to use state-of-the-art error-correcting codes. Low-density parity-check (LDPC) codes have become very popular in recent years due to their practical effectiveness under message-passing decoding. However, the analysis of LDPC codes is a difficult task. One approach was proposed in [8], and it is based on the consideration of so-called pseudocodewords and their pseudoweights. The approach was further explored in [3], [6]. In [1] and [2], the decoding of binary LDPC codes using linear-programming decoding was proposed, and the connections between linear-programming decoding and classical belief propagation decoding were established. Recently, pseudocodewords of non-binary codes were defined and some bounds on the pseudoweights were derived in [4].

In this work, we extend the approach in [2] towards coded modulation, in particular to codes over rings mapped to non-binary modulation signals. As was done in [2], we show that the problem of decoding may be formulated as a linear-programming (LP) problem for the non-binary case. We also show that an appropriate relaxation of the LP leads to a solution which has the ‘maximum likelihood (ML) certificate’ property, i.e. if the LP outputs a codeword, then it must be the ML codeword. Moreover, we show that if the LP output is integral, then it must correspond to the ML codeword. We define the graph-cover pseudocodewords of the code, and the LP pseudocodewords of the code, and prove the equivalence of these two concepts. This shows that the links between LP decoding on the relaxed polytope and message-passing decoding on the Tanner graph generalize to the non-binary case.

To demonstrate performance, LP decoding of the ternary Golay code is simulated, and the LP decoder is seen to perform approximately as well as codeword-error-rate optimum hard-decision decoding, and approximately 1.5 dB from the union bound for codeword-error-rate optimum soft-decision decoding.

We consider codes over finite rings (this includes codes over finite fields, but may be more general). Denote by R a ring with q elements, by 0 its additive identity, and let R -= R{0}. Let C be a linear [n, k] code with parity-check matrix H over R. The parity check matrix H has m ≥ nk rows.

Denote the set of column indices and the set of row indices of H by I = {1, 2, • • • , n} and J = {1, 2, • • • , m}, respectively. We use notation H j for the j-th row of H. Let the graph G = (V, E) be the Tanner graph of C associated with the matrix H, namely

and there is an edge between u i and v j if and only if H j,i = 0. We denote by N (v j ) the set of neighbors of the vertex v j , and by supp(c) the support of a vector c. Let

we associate the value c i with variable vertex u i for each i ∈ I. Parity-check j ∈ J is said to be satisfied if and only if i∈I H j,i • c i = 0. We say that the vector c is a codeword of the single parity-check code C j if and only if parity check j ∈ J is satisfied. Also, we say that the vector c is a codeword of C if and only if all parity checks j ∈ J are satisfied.

Definition 2.1:

) is a finite cover of the graph G = (V, E) if there exists a mapping Π : Ṽ → V which is a graph homomorphism (Π takes adjacent vertices of G to adjacent vertices of G), such that for every vertex v ∈ G and every ṽ ∈ Π -1 (v), the neighborhood N (ṽ) of ṽ is mapped bijectively to N (v).

Definition 2.2:

Fix some positive integer M . Let G = ( Ṽ, Ẽ) be an M -cover of the graph G = (V, E) representing the code C with parity-check matrix H. Denote the vertices in the sets Π -1 (u i ) and

respectively, where i ∈ I and j ∈ J .

Consider the linear code C of length M n over R, defined by the M m × M n parity-check matrix H. For 1 ≤ i * , j * ≤ M and i ∈ I, j ∈ J , we let i ′ = (i -1)M + i * , j ′ = (j -1)M + j * , and

Then, any vector p ∈ C has the form

We associate the value p i,ℓ ∈ R with the vertex u i,ℓ in

The word p ∈ C as above is called a graph-cover pseudocodeword of the code C. Sometimes, we consider the following n × q matrix representation, denoted P, of the pseudocodeword p:

Assume throughout that the codeword c = (c 1 , c2 , • • • , cn ) ∈ C has been transmitted over a q-ary input memoryless channel, and a corrupted word y = (y 1 , y 2 , • • • , y n ) ∈ Σ n has been received. Here Σ denotes the set of channel output symbols; we assume that this set either has finite cardinality, or is equal to R l or C l for some integer l ≥ 1. In practice, this channel may represent the combination of modulator and physical channel. We assume hereafter that all i