Best Known (7, 7+9, s)-Nets in Base 128
(7, 7+9, 387)-Net over F128 — Constructive and digital
Digital (7, 16, 387)-net over F128, using
- generalized (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 (0, 4, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 9, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 3, 129)-net over F128, using
(7, 7+9, 497)-Net over F128 — Digital
Digital (7, 16, 497)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 497, F128, 9) (dual of [497, 481, 10]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (1, 110 times 0) [i] based on linear OA(12815, 385, F128, 9) (dual of [385, 370, 10]-code), using
- construction XX applied to C1 = C([380,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([380,7]) [i] based on
- linear OA(12813, 381, F128, 8) (dual of [381, 368, 9]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12813, 381, F128, 8) (dual of [381, 368, 9]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12815, 381, F128, 9) (dual of [381, 366, 10]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12811, 381, F128, 7) (dual of [381, 370, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([380,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([380,7]) [i] based on
- 111 step Varšamov–Edel lengthening with (ri) = (1, 110 times 0) [i] based on linear OA(12815, 385, F128, 9) (dual of [385, 370, 10]-code), using
(7, 7+9, 515)-Net in Base 128 — Constructive
(7, 16, 515)-net in base 128, using
- base change [i] based on digital (5, 14, 515)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (1, 10, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (0, 4, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(7, 7+9, 546)-Net in Base 128
(7, 16, 546)-net in base 128, using
- base change [i] based on digital (5, 14, 546)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25614, 546, F256, 2, 9) (dual of [(546, 2), 1078, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2564, 257, F256, 2, 4) (dual of [(257, 2), 510, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;510,256) [i]
- linear OOA(25610, 289, F256, 2, 9) (dual of [(289, 2), 568, 10]-NRT-code), using
- extended algebraic-geometric NRT-code AGe(2;F,568P) [i] based on function field F/F256 with g(F) = 1 and N(F) ≥ 289, using
- linear OOA(2564, 257, F256, 2, 4) (dual of [(257, 2), 510, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25614, 546, F256, 2, 9) (dual of [(546, 2), 1078, 10]-NRT-code), using
(7, 7+9, 1390868)-Net in Base 128 — Upper bound on s
There is no (7, 16, 1390869)-net in base 128, because
- 1 times m-reduction [i] would yield (7, 15, 1390869)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 40 564905 188406 915756 623324 138413 > 12815 [i]