Best Known (43−18, 43, s)-Nets in Base 128
(43−18, 43, 1823)-Net over F128 — Constructive and digital
Digital (25, 43, 1823)-net over F128, using
- 1 times m-reduction [i] based on digital (25, 44, 1823)-net over F128, using
- net defined by OOA [i] based on linear OOA(12844, 1823, F128, 19, 19) (dual of [(1823, 19), 34593, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12844, 16408, F128, 19) (dual of [16408, 16364, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [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(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(12844, 16408, F128, 19) (dual of [16408, 16364, 20]-code), using
- net defined by OOA [i] based on linear OOA(12844, 1823, F128, 19, 19) (dual of [(1823, 19), 34593, 20]-NRT-code), using
(43−18, 43, 7282)-Net in Base 128 — Constructive
(25, 43, 7282)-net in base 128, using
- 1 times m-reduction [i] based on (25, 44, 7282)-net in base 128, using
- net defined by OOA [i] based on OOA(12844, 7282, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12844, 65539, S128, 19), using
- discarding factors based on OA(12844, 65542, S128, 19), using
- discarding parts of the base [i] based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding parts of the base [i] based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- discarding factors based on OA(12844, 65542, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12844, 65539, S128, 19), using
- net defined by OOA [i] based on OOA(12844, 7282, S128, 19, 19), using
(43−18, 43, 16410)-Net over F128 — Digital
Digital (25, 43, 16410)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12843, 16410, F128, 18) (dual of [16410, 16367, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(8) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1288, 26, F128, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(17) ⊂ Ce(8) [i] based on
(43−18, 43, large)-Net in Base 128 — Upper bound on s
There is no (25, 43, large)-net in base 128, because
- 16 times m-reduction [i] would yield (25, 27, large)-net in base 128, but