We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. In the fixed-degree regime, we prove that the coding rate stays bounded away from zero and that the relative distances on both sides stay bounded away from zero with probability tending to one as the blocklength grows. Moreover, within an explicit low-degree search window, we determine exactly which even regular degree choices in our construction attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.

翻译：我们从Hsu-Anastasopoulos码和MacKay-Neal码构造渐近好的嵌套Calderbank-Shor-Steane（CSS）码对。在固定度体制下，我们证明编码率保持有界远离零，且两侧的相对距离随码长增长以趋近于1的概率保持有界远离零。此外，在一个显式的低度搜索窗口内，我们精确确定了构造中哪些偶正则度选择能在两个组成侧均达到经典Gilbert-Varshamov（GV）界，进而达到固定有限度下的CSS GV界。