Best Known (20, 29, s)-Nets in Base 128
(20, 29, 524417)-Net over F128 — Constructive and digital
Digital (20, 29, 524417)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (16, 25, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- digital (0, 4, 129)-net over F128, using
(20, 29, 2097150)-Net in Base 128 — Constructive
(20, 29, 2097150)-net in base 128, using
- net defined by OOA [i] based on OOA(12829, 2097150, S128, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(12829, 8388601, S128, 9), using
- discarding factors based on OA(12829, large, S128, 9), using
- discarding parts of the base [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding parts of the base [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- discarding factors based on OA(12829, large, S128, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(12829, 8388601, S128, 9), using
(20, 29, 2097284)-Net over F128 — Digital
Digital (20, 29, 2097284)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12829, 2097284, F128, 9) (dual of [2097284, 2097255, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1284, 129, F128, 4) (dual of [129, 125, 5]-code or 129-arc in PG(3,128)), using
- extended Reed–Solomon code RSe(125,128) [i]
- linear OA(12825, 2097155, F128, 9) (dual of [2097155, 2097130, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- linear OA(1284, 129, F128, 4) (dual of [129, 125, 5]-code or 129-arc in PG(3,128)), using
- (u, u+v)-construction [i] based on
(20, 29, 4194300)-Net in Base 128
(20, 29, 4194300)-net in base 128, using
- net defined by OOA [i] based on OOA(12829, 4194300, S128, 12, 9), using
- OOA stacking with additional row [i] based on OOA(12829, 4194301, S128, 4, 9), using
- discarding parts of the base [i] based on linear OOA(25625, 4194301, F256, 4, 9) (dual of [(4194301, 4), 16777179, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25625, 4194301, F256, 2, 9) (dual of [(4194301, 2), 8388577, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25625, 8388602, F256, 9) (dual of [8388602, 8388577, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- OOA 2-folding [i] based on linear OA(25625, 8388602, F256, 9) (dual of [8388602, 8388577, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25625, 4194301, F256, 2, 9) (dual of [(4194301, 2), 8388577, 10]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(25625, 4194301, F256, 4, 9) (dual of [(4194301, 4), 16777179, 10]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12829, 4194301, S128, 4, 9), using
(20, 29, large)-Net in Base 128 — Upper bound on s
There is no (20, 29, large)-net in base 128, because
- 7 times m-reduction [i] would yield (20, 22, large)-net in base 128, but