Performance Comparisons

Table III demonstrates that ${\rm W}_{\Lambda _{24}}$ performs within $1$ dB of the distortion-rate function for rates in the range of 2-7 bits/sample. For this range, ${\rm W}_{\Lambda _{24}}$ outperforms many of the best quantizers in the literature, including 256-state trellis coded quantization (TCQ) [20], two-dimensional four-state trellis coded vector quantization (TCVQ) [26], Fischer's spherical vector quantization [14], and Lloyd-Max scalar quantization. With a large number of trellis states, TCQ and TCVQ may perhaps outperform ${\rm W}_{\Lambda _{24}}$; however, the reports of results in the literature have thus far been limited to trellises with 256 or fewer states because the design complexity of TCQ and TCVQ is somewhat prohibitive for larger trellises. Trellis-based scalar-vector quantization (TB-SVQ) [42] performs slightly better than ${\rm W}_{\Lambda _{24}}$ at a rate of $2$, but not at a rate of $3$.

Table III: Comparison of various quantization schemes for a memoryless Gaussian source. Values are listed as SNR in decibels. Blank entries indicate that referenced work does not contain a result. ${\rm W}_{\Lambda _{24}}$ is the proposed scheme using the Leech lattice as a shape codebook.

Method Rate:	1	2	3	4	5	6	7
Distortion-Rate function	6.02	12.04	18.06	24.08	30.10	36.12	42.14
${\rm W}_{\Lambda _{24}}$	2.44	11.02	17.36	23.33	29.29	35.27	41.33
TB-SVQ (4 state) [42]	5.39	11.18	16.92
TB-SVQ (32 state) [42]	5.49	11.28	17.05
Wilson (128 state) [43,20]	5.47	10.87	16.78
TCQ (256 state) [20]	5.56	11.04	16.64
TCVQ (2D, 16 state) [26]	5.29	10.84	16.62	22.63
Entropy coded scalar quantizer [22,20]	4.64	10.55	16.56	22.55	28.57	34.59	40.61
SVQ (estimated) [14]	4.49	10.51	16.53	22.55	28.57	34.59	40.61
GLA (kR=8) (simulation)		10.65		20.98
$Z^{16}$ lattice [16]		10.07	15.52	21.00	26.16	32.07	37.68
Unrestricted polar Quantizer [44]	4.40	9.63
Lloyd-Max Scalar [18,22]	4.40	9.30	14.62	20.22	26.02	31.89	37.81
Uniform scalar [17]	4.40	9.25	14.27	19.38	24.57	29.83	35.13