nisqec

Table of our constructed stabilizer matrices for NISQ Error Correction Codes. For CSS codes, "Hz" and "Hx" are binary parity-check matrices defining a CSS code. Minimum distances are calculated using the classical algorithm of [b], [c].

Table:

n	k	d	Average row weight	Row weight range	Construction	PCM	Paper(s)
144	12	12	8	8	GB_PK	PCM	[2], [4]
150	62	?	10	10	QT	PCM	[3]
168	76	4	12	12	GB_MMM	PCM	[3]
168	78	3	12	12	GQT	PCM	[3]
175	71	?	10	10	QT	PCM	[3]
180	26	6	6	6	GQT	PCM	[1]
180	26	7	8	8	GB_PK	PCM	[1]
180	28	5	6	6	QT	PCM	[1]
392	32	14	12	12	GB_PK	PCM	[2]
392	32	12	13	12 - 16	QT	PCM	[2]
392	48	12	13	12 - 16	QT	PCM	[2]
392	48	13	13	13	GB_MMM	PCM	[2]
480	40	12	12	12	GB_PK	PCM	[1]
500	42	4	8.1	6 - 12	QT	PCM	[1]
500	42	6	8.1	6 - 12	GQT	PCM	[1]
500	46	6	8.1	6 - 12	QT	PCM	[1]
500	188	4	10	10	QT	PCM	[3]
504	223	4	12	12	GQT	PCM	[3]
512	174	6	8	8	GB_MMM	PCM	[2], [3]
512	174	7	9	9	GB_MMM	PCM	[2], [3]
686	28	14	13	12 - 16	GQT	PCM	[1], [4]
686	34	13	13	12 - 16	QT	PCM	[1], [4]
688	30	$\geq 19$	16	16	GB_PK	PCM	[1], [4]

When citing, please refer to the paper introducing the code.

References.

[a]. D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse-graph codes for quantum error correction,” IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2315–2330, Oct. 2004.

[b] E. Rosnes and Ø. Ytrehus, “An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices,” IEEE Trans. Inf. Theory, vol. 55, no. 9, pp. 4167–4178, Sep. 2009.

[c] E. Rosnes, Ø. Ytrehus, M. A. Ambroze, and M. Tomlinson, “Addendum to “An efficient algorithm to find all small-size stopping sets of low- density parity-check matrices”, IEEE Trans. Inf. Theory, vol. 58, no. 1, pp. 164–171, Jan. 2012.

[1] O. Å. Mostad, E. Rosnes and H.-Y. Lin, "Asymptotically Good Generalized Quantum Tanner Codes," in IEEE Journal on Selected Areas in Information Theory, vol. 6, pp. 367-382, 2025, doi: 10.1109/JSAIT.2025.3594310.

[2]. O. Å. Mostad, H.-Y. Lin, E. Rosnes, D.-S. Lee, and C.-Y. Lai, "Advancing Finite-Length Quantum Error Correction Using Generalized Bicycle Codes," 2025 13th International Symposium on Topics in Coding (ISTC), Los Angeles, CA, USA, 2025, pp. 1-5, doi: 10.1109/ISTC65386.2025.11154497, arXiv:2505.06157v1 [quant-ph].

[3]. O. Å. Mostad, E. Rosnes, and H.-Y. Lin, "Improved Construction of Generalized Quantum Tanner Codes," 2025 13th International Symposium on Topics in Coding (ISTC), Los Angeles, CA, USA, 2025, pp. 1-5, doi: 10.1109/ISTC65386.2025.11154489.

[4]. O. Å. Mostad, E. Rosnes, and H.-Y. Lin, "Improved Decoding of Quantum Tanner Codes Using Generalized Check Nodes," March 2026, arXiv:2603.05486v1 [quant-ph].