A Novel Scheme of Product Polar Codes with High Efficiency and Rate Compatibility

Abstract

This paper presents a new structure of product codes formed by combining two polar codes. The encoding performance of these codes is verified by implementing an exhaustive search algorithm, which determines their minimum weight specifications. Conducted analysis and simulations confirm that with the equal code length and rate, the newly proposed codes outperform the conventional polar codes in high energy per bit-to-noise ratios (Eb/N0≤5 dB). This is concluded from punctured and nonpunctured product codes.

A Modified Approach to Punctured Product Polar Codes

“…These specifications deteriorate performance of codes at medium to high signal-to-noise ratios, which is mainly dominated by their minimum weight specifications. By contrast, with the same code rate and length, it is possible to construct product codes whose minimum weights have lower multiplicity than those obtained from single polar codes [8].…”

“…Indeed, all 8 codewords expressing columns of the matrix are punctured by two bits. This sums up a (48, 16) product polar code formed by combining (8,4,4) and (6,4,4) polar codes.…”

“…Hence, a two-dimensional codeword matrix (x) at the output of inner encoder is obtained. One may observe that only one bit from l −1 = 7 codewords of inner (8,4,4) code is punctured. Again, for this code, the first bit of the codeword is the most unreliable one.…”

“…In general, performance of SPCs is limited due to the existence of a low minimum weight, or a relatively high minimum weight with high multiplicity. As a solution, product polar codes (PPCs) were proposed, which are constructed by the serial combination of two half-rate polar codes [8]. These codes generate the same minimum weight but with a lower multiplicity compared to SPCs with rate 1.…”

“…In the proposed method, puncturing of the product code is only conducted at one of its constituent codes. Low performance of PPCs in the waterfall region is mainly related to the weak performance of their punctured constituent codes [8]. This paper presents an algorithm, which improves the performance of punctured product polar codes, where their two constituent codes are punctured.…”