Best Known (40−12, 40, s)-Nets in Base 128
(40−12, 40, 349654)-Net over F128 — Constructive and digital
Digital (28, 40, 349654)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (22, 34, 349525)-net over F128, using
- net defined by OOA [i] based on linear OOA(12834, 349525, F128, 12, 12) (dual of [(349525, 12), 4194266, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12834, 2097150, F128, 12) (dual of [2097150, 2097116, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12834, 2097150, F128, 12) (dual of [2097150, 2097116, 13]-code), using
- net defined by OOA [i] based on linear OOA(12834, 349525, F128, 12, 12) (dual of [(349525, 12), 4194266, 13]-NRT-code), using
- digital (0, 6, 129)-net over F128, using
(40−12, 40, 1398100)-Net in Base 128 — Constructive
(28, 40, 1398100)-net in base 128, using
- base change [i] based on digital (23, 35, 1398100)-net over F256, using
- 2561 times duplication [i] based on digital (22, 34, 1398100)-net over F256, using
- net defined by OOA [i] based on linear OOA(25634, 1398100, F256, 12, 12) (dual of [(1398100, 12), 16777166, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(25634, 8388600, F256, 12) (dual of [8388600, 8388566, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(25634, 8388600, F256, 12) (dual of [8388600, 8388566, 13]-code), using
- net defined by OOA [i] based on linear OOA(25634, 1398100, F256, 12, 12) (dual of [(1398100, 12), 16777166, 13]-NRT-code), using
- 2561 times duplication [i] based on digital (22, 34, 1398100)-net over F256, using
(40−12, 40, 2097284)-Net over F128 — Digital
Digital (28, 40, 2097284)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12840, 2097284, F128, 12) (dual of [2097284, 2097244, 13]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1286, 129, F128, 6) (dual of [129, 123, 7]-code or 129-arc in PG(5,128)), using
- extended Reed–Solomon code RSe(123,128) [i]
- linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- linear OA(1286, 129, F128, 6) (dual of [129, 123, 7]-code or 129-arc in PG(5,128)), using
- (u, u+v)-construction [i] based on
(40−12, 40, 4194301)-Net in Base 128
(28, 40, 4194301)-net in base 128, using
- base change [i] based on digital (23, 35, 4194301)-net over F256, using
- 2561 times duplication [i] based on digital (22, 34, 4194301)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- 2561 times duplication [i] based on digital (22, 34, 4194301)-net over F256, using
(40−12, 40, large)-Net in Base 128 — Upper bound on s
There is no (28, 40, large)-net in base 128, because
- 10 times m-reduction [i] would yield (28, 30, large)-net in base 128, but