Best Known (51, 51+26, s)-Nets in Base 128
(51, 51+26, 161319)-Net over F128 — Constructive and digital
Digital (51, 77, 161319)-net over F128, using
- 1281 times duplication [i] based on digital (50, 76, 161319)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 161319, F128, 26, 26) (dual of [(161319, 26), 4194218, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(12876, 2097147, F128, 26) (dual of [2097147, 2097071, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(12876, 2097147, F128, 26) (dual of [2097147, 2097071, 27]-code), using
- net defined by OOA [i] based on linear OOA(12876, 161319, F128, 26, 26) (dual of [(161319, 26), 4194218, 27]-NRT-code), using
(51, 51+26, 699053)-Net over F128 — Digital
Digital (51, 77, 699053)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12877, 699053, F128, 3, 26) (dual of [(699053, 3), 2097082, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12877, 2097159, F128, 26) (dual of [2097159, 2097082, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- OOA 3-folding [i] based on linear OA(12877, 2097159, F128, 26) (dual of [2097159, 2097082, 27]-code), using
(51, 51+26, large)-Net in Base 128 — Upper bound on s
There is no (51, 77, large)-net in base 128, because
- 24 times m-reduction [i] would yield (51, 53, large)-net in base 128, but