Best Known (15, 15+7, s)-Nets in Base 128
(15, 15+7, 699180)-Net over F128 — Constructive and digital
Digital (15, 22, 699180)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (12, 19, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- digital (0, 3, 129)-net over F128, using
(15, 15+7, 2097285)-Net over F128 — Digital
Digital (15, 22, 2097285)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12822, 2097285, F128, 7) (dual of [2097285, 2097263, 8]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1283, 130, F128, 3) (dual of [130, 127, 4]-code or 130-arc in PG(2,128) or 130-cap in PG(2,128)), using
- linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- (u, u+v)-construction [i] based on
(15, 15+7, 2796200)-Net in Base 128 — Constructive
(15, 22, 2796200)-net in base 128, using
- net defined by OOA [i] based on OOA(12822, 2796200, S128, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12822, 8388601, S128, 7), using
- discarding factors based on OA(12822, large, S128, 7), using
- discarding parts of the base [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding parts of the base [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- discarding factors based on OA(12822, large, S128, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12822, 8388601, S128, 7), using
(15, 15+7, 4774850)-Net in Base 128
(15, 22, 4774850)-net in base 128, using
- net defined by OOA [i] based on OOA(12822, 4774850, S128, 9, 7), using
- OOA stacking with additional row [i] based on OOA(12822, 4774851, S128, 3, 7), using
- discarding parts of the base [i] based on linear OOA(25619, 4774851, F256, 3, 7) (dual of [(4774851, 3), 14324534, 8]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25619, 4774851, F256, 7) (dual of [4774851, 4774832, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25619, 4774851, F256, 7) (dual of [4774851, 4774832, 8]-code), using
- discarding parts of the base [i] based on linear OOA(25619, 4774851, F256, 3, 7) (dual of [(4774851, 3), 14324534, 8]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12822, 4774851, S128, 3, 7), using
(15, 15+7, large)-Net in Base 128 — Upper bound on s
There is no (15, 22, large)-net in base 128, because
- 5 times m-reduction [i] would yield (15, 17, large)-net in base 128, but