Best Known (64−26, 64, s)-Nets in Base 128
(64−26, 64, 1389)-Net over F128 — Constructive and digital
Digital (38, 64, 1389)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (25, 51, 1260)-net over F128, using
- net defined by OOA [i] based on linear OOA(12851, 1260, F128, 26, 26) (dual of [(1260, 26), 32709, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(12851, 16380, F128, 26) (dual of [16380, 16329, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(12851, 16380, F128, 26) (dual of [16380, 16329, 27]-code), using
- net defined by OOA [i] based on linear OOA(12851, 1260, F128, 26, 26) (dual of [(1260, 26), 32709, 27]-NRT-code), using
- digital (0, 13, 129)-net over F128, using
(64−26, 64, 5042)-Net in Base 128 — Constructive
(38, 64, 5042)-net in base 128, using
- t-expansion [i] based on (37, 64, 5042)-net in base 128, using
- base change [i] based on digital (29, 56, 5042)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- base change [i] based on digital (29, 56, 5042)-net over F256, using
(64−26, 64, 19886)-Net over F128 — Digital
Digital (38, 64, 19886)-net over F128, using
(64−26, 64, 21851)-Net in Base 128
(38, 64, 21851)-net in base 128, using
- base change [i] based on digital (30, 56, 21851)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25656, 21851, F256, 3, 26) (dual of [(21851, 3), 65497, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25656, 65553, F256, 26) (dual of [65553, 65497, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(25656, 65553, F256, 26) (dual of [65553, 65497, 27]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25656, 21851, F256, 3, 26) (dual of [(21851, 3), 65497, 27]-NRT-code), using
(64−26, 64, large)-Net in Base 128 — Upper bound on s
There is no (38, 64, large)-net in base 128, because
- 24 times m-reduction [i] would yield (38, 40, large)-net in base 128, but