Best Known (63−28, 63, s)-Nets in Base 128
(63−28, 63, 1172)-Net over F128 — Constructive and digital
Digital (35, 63, 1172)-net over F128, using
- net defined by OOA [i] based on linear OOA(12863, 1172, F128, 28, 28) (dual of [(1172, 28), 32753, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(12863, 16408, F128, 28) (dual of [16408, 16345, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(12863, 16410, F128, 28) (dual of [16410, 16347, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(18) [i] based on
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1288, 26, F128, 8) (dual of [26, 18, 9]-code or 26-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(27) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12863, 16410, F128, 28) (dual of [16410, 16347, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(12863, 16408, F128, 28) (dual of [16408, 16345, 29]-code), using
(63−28, 63, 4681)-Net in Base 128 — Constructive
(35, 63, 4681)-net in base 128, using
- net defined by OOA [i] based on OOA(12863, 4681, S128, 28, 28), using
- OA 14-folding and stacking [i] based on OA(12863, 65534, S128, 28), using
- discarding factors based on OA(12863, 65538, S128, 28), using
- discarding parts of the base [i] based on linear OA(25655, 65538, F256, 28) (dual of [65538, 65483, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- discarding parts of the base [i] based on linear OA(25655, 65538, F256, 28) (dual of [65538, 65483, 29]-code), using
- discarding factors based on OA(12863, 65538, S128, 28), using
- OA 14-folding and stacking [i] based on OA(12863, 65534, S128, 28), using
(63−28, 63, 8786)-Net over F128 — Digital
Digital (35, 63, 8786)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12863, 8786, F128, 28) (dual of [8786, 8723, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(12863, 16410, F128, 28) (dual of [16410, 16347, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(18) [i] based on
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1288, 26, F128, 8) (dual of [26, 18, 9]-code or 26-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(27) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12863, 16410, F128, 28) (dual of [16410, 16347, 29]-code), using
(63−28, 63, large)-Net in Base 128 — Upper bound on s
There is no (35, 63, large)-net in base 128, because
- 26 times m-reduction [i] would yield (35, 37, large)-net in base 128, but