Best Known (125, 125+24, s)-Nets in Base 8
(125, 125+24, 174763)-Net over F8 — Constructive and digital
Digital (125, 149, 174763)-net over F8, using
- t-expansion [i] based on digital (124, 149, 174763)-net over F8, using
- net defined by OOA [i] based on linear OOA(8149, 174763, F8, 25, 25) (dual of [(174763, 25), 4368926, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8149, 2097157, F8, 25) (dual of [2097157, 2097008, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, 2097160, F8, 25) (dual of [2097160, 2097011, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8149, 2097160, F8, 25) (dual of [2097160, 2097011, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8149, 2097157, F8, 25) (dual of [2097157, 2097008, 26]-code), using
- net defined by OOA [i] based on linear OOA(8149, 174763, F8, 25, 25) (dual of [(174763, 25), 4368926, 26]-NRT-code), using
(125, 125+24, 1538452)-Net over F8 — Digital
Digital (125, 149, 1538452)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8149, 1538452, F8, 24) (dual of [1538452, 1538303, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, 2097167, F8, 24) (dual of [2097167, 2097018, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8134, 2097152, F8, 22) (dual of [2097152, 2097018, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8149, 2097167, F8, 24) (dual of [2097167, 2097018, 25]-code), using
(125, 125+24, large)-Net in Base 8 — Upper bound on s
There is no (125, 149, large)-net in base 8, because
- 22 times m-reduction [i] would yield (125, 127, large)-net in base 8, but