Best Known (54, 54+24, s)-Nets in Base 128
(54, 54+24, 174765)-Net over F128 — Constructive and digital
Digital (54, 78, 174765)-net over F128, using
- 1281 times duplication [i] based on digital (53, 77, 174765)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 174765, F128, 24, 24) (dual of [(174765, 24), 4194283, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12877, 2097180, F128, 24) (dual of [2097180, 2097103, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, 2097183, F128, 24) (dual of [2097183, 2097106, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- 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(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(12877, 2097183, F128, 24) (dual of [2097183, 2097106, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12877, 2097180, F128, 24) (dual of [2097180, 2097103, 25]-code), using
- net defined by OOA [i] based on linear OOA(12877, 174765, F128, 24, 24) (dual of [(174765, 24), 4194283, 25]-NRT-code), using
(54, 54+24, 1691537)-Net over F128 — Digital
Digital (54, 78, 1691537)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12878, 1691537, F128, 24) (dual of [1691537, 1691459, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12878, 2097187, F128, 24) (dual of [2097187, 2097109, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(14) [i] based on
- 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(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1288, 35, F128, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(23) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(12878, 2097187, F128, 24) (dual of [2097187, 2097109, 25]-code), using
(54, 54+24, large)-Net in Base 128 — Upper bound on s
There is no (54, 78, large)-net in base 128, because
- 22 times m-reduction [i] would yield (54, 56, large)-net in base 128, but