Best Known (30, 30+13, s)-Nets in Base 128
(30, 30+13, 349654)-Net over F128 — Constructive and digital
Digital (30, 43, 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 (24, 37, 349525)-net over F128, using
- net defined by OOA [i] based on linear OOA(12837, 349525, F128, 13, 13) (dual of [(349525, 13), 4543788, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12837, 2097151, F128, 13) (dual of [2097151, 2097114, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12837, 2097151, F128, 13) (dual of [2097151, 2097114, 14]-code), using
- net defined by OOA [i] based on linear OOA(12837, 349525, F128, 13, 13) (dual of [(349525, 13), 4543788, 14]-NRT-code), using
- digital (0, 6, 129)-net over F128, using
(30, 30+13, 1398100)-Net in Base 128 — Constructive
(30, 43, 1398100)-net in base 128, using
- net defined by OOA [i] based on OOA(12843, 1398100, S128, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12843, 8388601, S128, 13), using
- discarding factors based on OA(12843, large, S128, 13), using
- discarding parts of the base [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding parts of the base [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- discarding factors based on OA(12843, large, S128, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12843, 8388601, S128, 13), using
(30, 30+13, 2097284)-Net over F128 — Digital
Digital (30, 43, 2097284)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12843, 2097284, F128, 13) (dual of [2097284, 2097241, 14]-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(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(11) [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
(30, 30+13, large)-Net in Base 128 — Upper bound on s
There is no (30, 43, large)-net in base 128, because
- 11 times m-reduction [i] would yield (30, 32, large)-net in base 128, but