Best Known (24, 24+19, s)-Nets in Base 128
(24, 24+19, 1822)-Net over F128 — Constructive and digital
Digital (24, 43, 1822)-net over F128, using
- 1281 times duplication [i] based on digital (23, 42, 1822)-net over F128, using
- net defined by OOA [i] based on linear OOA(12842, 1822, F128, 19, 19) (dual of [(1822, 19), 34576, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12842, 16399, F128, 19) (dual of [16399, 16357, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12842, 16402, F128, 19) (dual of [16402, 16360, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12842, 16402, F128, 19) (dual of [16402, 16360, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12842, 16399, F128, 19) (dual of [16399, 16357, 20]-code), using
- net defined by OOA [i] based on linear OOA(12842, 1822, F128, 19, 19) (dual of [(1822, 19), 34576, 20]-NRT-code), using
(24, 24+19, 7281)-Net in Base 128 — Constructive
(24, 43, 7281)-net in base 128, using
- net defined by OOA [i] based on OOA(12843, 7281, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12843, 65530, S128, 19), using
- discarding factors based on OA(12843, 65538, S128, 19), using
- discarding parts of the base [i] based on linear OA(25637, 65538, F256, 19) (dual of [65538, 65501, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(25637, 65536, F256, 19) (dual of [65536, 65499, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(25637, 65538, F256, 19) (dual of [65538, 65501, 20]-code), using
- discarding factors based on OA(12843, 65538, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12843, 65530, S128, 19), using
(24, 24+19, 9075)-Net over F128 — Digital
Digital (24, 43, 9075)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12843, 9075, F128, 19) (dual of [9075, 9032, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12843, 16404, F128, 19) (dual of [16404, 16361, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- 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(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12843, 16404, F128, 19) (dual of [16404, 16361, 20]-code), using
(24, 24+19, large)-Net in Base 128 — Upper bound on s
There is no (24, 43, large)-net in base 128, because
- 17 times m-reduction [i] would yield (24, 26, large)-net in base 128, but