Best Known (63−31, 63, s)-Nets in Base 128
(63−31, 63, 1092)-Net over F128 — Constructive and digital
Digital (32, 63, 1092)-net over F128, using
- 1282 times duplication [i] based on digital (30, 61, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
(63−31, 63, 4130)-Net over F128 — Digital
Digital (32, 63, 4130)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12863, 4130, F128, 3, 31) (dual of [(4130, 3), 12327, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12863, 5464, F128, 3, 31) (dual of [(5464, 3), 16329, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12863, 16392, F128, 31) (dual of [16392, 16329, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- OOA 3-folding [i] based on linear OA(12863, 16392, F128, 31) (dual of [16392, 16329, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(12863, 5464, F128, 3, 31) (dual of [(5464, 3), 16329, 32]-NRT-code), using
(63−31, 63, large)-Net in Base 128 — Upper bound on s
There is no (32, 63, large)-net in base 128, because
- 29 times m-reduction [i] would yield (32, 34, large)-net in base 128, but