Best Known (28, 28+18, s)-Nets in Base 128
(28, 28+18, 1971)-Net over F128 — Constructive and digital
Digital (28, 46, 1971)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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 (18, 36, 1821)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 1821, F128, 18, 18) (dual of [(1821, 18), 32742, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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 Ce(17) ⊂ Ce(15) [i] based on
- OA 9-folding and stacking [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- net defined by OOA [i] based on linear OOA(12836, 1821, F128, 18, 18) (dual of [(1821, 18), 32742, 19]-NRT-code), using
- digital (1, 10, 150)-net over F128, using
(28, 28+18, 7283)-Net in Base 128 — Constructive
(28, 46, 7283)-net in base 128, using
- t-expansion [i] based on (27, 46, 7283)-net in base 128, using
- net defined by OOA [i] based on OOA(12846, 7283, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12846, 65548, S128, 19), using
- discarding parts of the base [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [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(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,9]) ⊂ C([0,7]) [i] based on
- discarding parts of the base [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on OA(12846, 65548, S128, 19), using
- net defined by OOA [i] based on OOA(12846, 7283, S128, 19, 19), using
(28, 28+18, 28453)-Net over F128 — Digital
Digital (28, 46, 28453)-net over F128, using
(28, 28+18, large)-Net in Base 128 — Upper bound on s
There is no (28, 46, large)-net in base 128, because
- 16 times m-reduction [i] would yield (28, 30, large)-net in base 128, but