Short Turbo Codes over High Order Fields

Coding in the Finite-Blocklength Regime: Bounds Based on Laplace Integrals and Their Asymptotic Approximations

“…Although the technique used in this paper is not able to deal with the RCU achievability bound, use of the weaker κβ achievability bound of [9,Theorem 25] clearly reveals the 1 dB gap existing between what we could achieve and the performance of a standard message passing decoder, the gap being valid over a very large blocklength range (i.e., for short as well as for long codes). It is anyway worth mentioning that for very short packet sizes, one can construct codes (and decoders) that approach the achievability bound, and in some regions (high error rate) even beat it [22]. For the BSC we are finally providing reliable approximations to both the PPV metaconverse and the RCU achievability bounds, thus improving over the results available in [9] and based upon a series expression.…”

“…The codes used in figure are both rate R = 1 2 , the shorter one being the (k, n) = (1320, 2640) Margulis code [27], the longer one being taken from [28]. It is in any case worth recalling that, for very short packet sizes one can construct codes (and decoders) that approach the achievability bound, and in some regions (high error rate) even beat it [22].…”

“…We are therefore interested in a counterpart to (3) where Λ is replaced by Λ ′ . With a little effort, and by exploiting the properties of a non-central chi-squared random variable [31, §2.2.3] and the same method used in [14], we obtain (22). Note that in (22) the threshold level λ ′ to be used in connection with Λ ′ has been replaced by λ since, operatively, they have the same meaning.…”

Jointly optimized multiple Reed-Muller codes for wireless half-duplex coded-cooperative network with joint decoding

“…The code rate at the source is R c = 1/2, and the overall code rate of the coded-cooperative scheme is R 0 c ¼ 1=4. For case II, the non-cooperative scheme consists of two encoders C 1 = ℛ(2, 5) and C 2 = ℛ(1, 5) at the source node, and the optimum bit selection rule for the second encoder is λ o = λ 1 also shown in (20). The direct sum of two codewords generated by the two encoders is determined as u + v, and concatenated with the codeword u generated by the first encoder C 1 = ℛ(2, 5), to construct a codeword |u|u + v|, which is then BPSKmodulated and sent to the destination.…”

“…In order to achieve the coded-cooperative diversity, many distributed coding schemes have been reported in the literature such as the convolutional codes [10,11], distributed space-time coding [12,13], distributed low-density parity-check codes (D-LDPC) [14,15] and distributed turbo codes (DTC) [16,17] and more recently the polar codes [18,19]. However, the BER performance of binary turbo and the LDPC codes largely depends on the information block size, the longer the better and vice versa, whereas for the short non-binary turbo and LDPC codes, reasonable performances under the iterative decoding are reported in [20,21]. In many existing and emerging applications (such as device-todevice and sensor networks), it is possible to have scenarios, which may transmit small information block size.…”