Best Known (50−15, 50, s)-Nets in Base 128
(50−15, 50, 299722)-Net over F128 — Constructive and digital
Digital (35, 50, 299722)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (28, 43, 299593)-net over F128, using
- net defined by OOA [i] based on linear OOA(12843, 299593, F128, 15, 15) (dual of [(299593, 15), 4493852, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using
- net defined by OOA [i] based on linear OOA(12843, 299593, F128, 15, 15) (dual of [(299593, 15), 4493852, 16]-NRT-code), using
- digital (0, 7, 129)-net over F128, using
(50−15, 50, 1198371)-Net in Base 128 — Constructive
(35, 50, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- discarding factors based on OA(12850, large, S128, 15), using
- discarding parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- discarding factors based on OA(12850, large, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
(50−15, 50, 2097284)-Net over F128 — Digital
Digital (35, 50, 2097284)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12850, 2097284, F128, 15) (dual of [2097284, 2097234, 16]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1287, 129, F128, 7) (dual of [129, 122, 8]-code or 129-arc in PG(6,128)), using
- extended Reed–Solomon code RSe(122,128) [i]
- the expurgated narrow-sense BCH-code C(I) with length 129 | 1282−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12843, 2097155, F128, 15) (dual of [2097155, 2097112, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(1287, 129, F128, 7) (dual of [129, 122, 8]-code or 129-arc in PG(6,128)), using
- (u, u+v)-construction [i] based on
(50−15, 50, large)-Net in Base 128 — Upper bound on s
There is no (35, 50, large)-net in base 128, because
- 13 times m-reduction [i] would yield (35, 37, large)-net in base 128, but