Best Known (40, 40+31, s)-Nets in Base 128
(40, 40+31, 1094)-Net over F128 — Constructive and digital
Digital (40, 71, 1094)-net over F128, using
- 1281 times duplication [i] based on digital (39, 70, 1094)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 1094, F128, 31, 31) (dual of [(1094, 31), 33844, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12870, 16411, F128, 31) (dual of [16411, 16341, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16414, F128, 31) (dual of [16414, 16344, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1289, 29, F128, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16414, F128, 31) (dual of [16414, 16344, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12870, 16411, F128, 31) (dual of [16411, 16341, 32]-code), using
- net defined by OOA [i] based on linear OOA(12870, 1094, F128, 31, 31) (dual of [(1094, 31), 33844, 32]-NRT-code), using
(40, 40+31, 4369)-Net in Base 128 — Constructive
(40, 71, 4369)-net in base 128, using
- 1281 times duplication [i] based on (39, 70, 4369)-net in base 128, using
- net defined by OOA [i] based on OOA(12870, 4369, S128, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(12870, 65536, S128, 31), using
- discarding factors based on OA(12870, 65538, S128, 31), using
- discarding parts of the base [i] based on linear OA(25661, 65538, F256, 31) (dual of [65538, 65477, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- discarding parts of the base [i] based on linear OA(25661, 65538, F256, 31) (dual of [65538, 65477, 32]-code), using
- discarding factors based on OA(12870, 65538, S128, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(12870, 65536, S128, 31), using
- net defined by OOA [i] based on OOA(12870, 4369, S128, 31, 31), using
(40, 40+31, 11198)-Net over F128 — Digital
Digital (40, 71, 11198)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12871, 11198, F128, 31) (dual of [11198, 11127, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12871, 16416, F128, 31) (dual of [16416, 16345, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(19) [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(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12810, 32, F128, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(30) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12871, 16416, F128, 31) (dual of [16416, 16345, 32]-code), using
(40, 40+31, large)-Net in Base 128 — Upper bound on s
There is no (40, 71, large)-net in base 128, because
- 29 times m-reduction [i] would yield (40, 42, large)-net in base 128, but