Turbo product decoding of cubic tensor codes

TL;DR

This work addresses the need for long, low-rate FEC codes that remain practical to decode by extending turbo product codes to cubic tensor codes with dimensions $(n^3,k^3)$ and rate $R=(k/n)^3$. It leverages accurate soft-output decoders, such as GRAND-based SISO decoders, to enable highly parallel iterative decoding across the three dimensions (rows, columns, tubes), achieving competitive performance with LDPCs while requiring few iterations. The paper provides the formalism for 3D tensor code construction, analyzes encoding complexity and design space, and identifies hyperparameters—most notably $oldsymbol{ }=0.7$ for cubic codes—that yield favorable BER/BLER performance. Empirical results on AWGN/BPSK demonstrate strong error-rate performance with modest iteration counts, confirming cubic tensor codes as a flexible platform for powerful, low-rate, highly parallelizable FEC suitable for challenging environments and hardware implementations.

Abstract

Long, powerful soft detection forward error correction codes are typically constructed by concatenation of shorter component codes that are decoded through iterative Soft-Input Soft-Output (SISO) procedures. The current gold-standard is Low Density Parity Check (LDPC) codes, which are built from weak single parity check component codes that are capable of producing accurate SO. Due to the recent development of SISO decoders that produce highly accurate SO with codes that have multiple redundant bits, square product code constructions that can avail of more powerful component codes have been shown to be competitive with the LDPC codes in the 5G New Radio standard in terms of decoding performance while requiring fewer iterations to converge. Motivated by applications that require more powerful low-rate codes, in the present paper we explore the possibility of extending this design space by considering the construction and decoding of cubic tensor codes.

Figures (9)

• Figure 1: Block error rate (solid) and bit error rate (dashed) vs SNR of a $[15,10]$ component, a $[225,100]$ product, and a $[3375,1000]$ cubic code constructed with a CRC component code with polynomial 0x15 in Koopman notation koopman2009cyclic. Component code decoding is performed with 1-line ORBGRAND and iteratively decoded with a SISO version for the product and cubic code.
• Figure 2: A cubic tensor code of dimension $(n^3,k^3)$, where each row, column and tube forms an $(n,k)$ codeword, for a code rate of $(k/n)^3$.
• Figure 3: For moderate redundancy component codes with $25$ or fewer redundant bits, available lengths and rates for component, square tensor product, and cubic tensor product codes with components of the same dimensions.
• Figure 4: Performance of a [256,121] product code constructed using a [16,11] eBCH component code in an AWGN channel where iterative decoding is performed using SISO GRAND adapted for square product codes for different $\alpha$ values. Throughout decoding, a maximum of 20 iterations are considered with a list size L = 4 or an a posteriori list-BLER $<10^{-5}$ is reached.
• Figure 5: Performance of a [4096,1331] cubic tensor code constructed using a [16,11] eBCH component code in an AWGN channel where iterative decoding is performed using SISO GRAND adapted for cubic tensor codes for different $\alpha$ values. Throughout decoding, a maximum of 30 iterations are considered with a list size L = 4 or an a posteriori list-BLER $<10^{-5}$ is reached.