Best Known (23, 23+15, s)-Nets in Base 128
(23, 23+15, 2491)-Net over F128 — Constructive and digital
Digital (23, 38, 2491)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (15, 30, 2341)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- digital (1, 8, 150)-net over F128, using
(23, 23+15, 9364)-Net in Base 128 — Constructive
(23, 38, 9364)-net in base 128, using
- net defined by OOA [i] based on OOA(12838, 9364, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12838, 65549, S128, 15), using
- discarding factors based on OA(12838, 65550, S128, 15), using
- discarding parts of the base [i] based on linear OA(25633, 65550, F256, 15) (dual of [65550, 65517, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- discarding parts of the base [i] based on linear OA(25633, 65550, F256, 15) (dual of [65550, 65517, 16]-code), using
- discarding factors based on OA(12838, 65550, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12838, 65549, S128, 15), using
(23, 23+15, 24966)-Net over F128 — Digital
Digital (23, 38, 24966)-net over F128, using
(23, 23+15, large)-Net in Base 128 — Upper bound on s
There is no (23, 38, large)-net in base 128, because
- 13 times m-reduction [i] would yield (23, 25, large)-net in base 128, but