Application to VDSL2

Turbo trellis coded modulations (TTCM) [4] have been proposed for channel coding of VDSL2 using QAM modulations for a large range of spectral efficiency values [8]. The TTCM is serially concatenated with an outer code. The latter is a standard Reed-Solomon (RS) of length 255 and dimension 239 defined over the field $GF(256)$ . As designed in [8], a 128-QAM constellation is adequately partitioned and labelled, two bits in a QAM symbol label are precoded with a rate $1/2$ binary parallel turbo code yielding a final spectral efficiency of 5 bits/sec/Hz. After taking into account the RS coding rate, the information rate of RS+TTCM is $R=239/255 \times 5=2.34$ bits per real dimension. The TTCM block length is 1022 QAM symbols (Turbo interleaver of size 2044 bits). Hence, the code is in a real space of dimension n=2044.
Let us compare the performance of the RS+TTCM to the error rate of an optimal code having the same parameters. In order to convert the word error probability $P_{ew}$ given by (2) into a bit error probability $P_{eb}$ , we propose the following: Assume that a codeword on the n-dimensional sphere is surrounded by $\tau_n$ neighbours and assume that decoding errors yield only one of those neighbours. Each codeword is labelled by $nR$ bits. The considered neighbours are labelled by $\log_2(\tau_n)$ bits, suppose that $\log_2(\tau_n) < nR$ . If random binary labelling is used to index the $\tau_n$ neighbours, then we have

$$P_{eb} ~\approx~ \frac{\frac{1}{2}\log_2(\tau_n)}{nR} ~P_{ew}$$ (3)

Let $\tau^*_n$ denote the greatest value attainded by the kissing number of an n-dimensional sphere packing. It is known that [5]

$$2^{0.2075n(1+o(1))} ~\le~ \tau^*_n ~\le~ 2^{0.401n(1+o(1))}$$ (4)

The lower bound has been proved by Kabatiansky and Levenshtein [3] and the upper bound by Wyner [2]. Finally by using the right inequality in (4) we get

$$P_{eb} ~\lesssim~ \frac{0.401}{2R} ~P_{ew}$$ (5)

Figure 5 illustrates the bit error rate of RS+TTCM versus optimal codes at finite length. The coding gain gap is about 2.45 dB (3dB from capacity limit at $n=+\infty$ ). The capacity limit is given by the rate-distortion bound

$$P_{eb}(n=+\infty) ~\ge~ H_2^{-1}\left( 1-\frac{C}{R}\right)$$ (6)

Since $R$ is the rate per real dimension, then $C=\frac{1}{2}\log_2(1+2R\frac{E_b}{N_0})$ . Multilevel coded modulations with multistage decoding [6] exhibit performance similar to those of RS+TTCM. We believe that feasible coded modulations exist at less than 1dB from optimal codes at such high information rates.