Best Known (53, 53+24, s)-Nets in Base 128
(53, 53+24, 174765)-Net over F128 — Constructive and digital
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
(53, 53+24, 1356746)-Net over F128 — Digital
Digital (53, 77, 1356746)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12877, 1356746, F128, 24) (dual of [1356746, 1356669, 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
(53, 53+24, large)-Net in Base 128 — Upper bound on s
There is no (53, 77, large)-net in base 128, because
- 22 times m-reduction [i] would yield (53, 55, large)-net in base 128, but