Best Known (15−9, 15, s)-Nets in Base 128
(15−9, 15, 300)-Net over F128 — Constructive and digital
Digital (6, 15, 300)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (1, 10, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128 (see above)
- digital (1, 5, 150)-net over F128, using
(15−9, 15, 385)-Net over F128 — Digital
Digital (6, 15, 385)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [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
(15−9, 15, 513)-Net in Base 128 — Constructive
(6, 15, 513)-net in base 128, using
- net defined by OOA [i] based on OOA(12815, 513, S128, 12, 9), using
- OOA stacking with additional row [i] based on OOA(12815, 514, S128, 4, 9), using
- discarding parts of the base [i] based on linear OOA(25613, 514, F256, 4, 9) (dual of [(514, 4), 2043, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2564, 257, F256, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1024,256) [i]
- linear OOA(2569, 257, F256, 4, 9) (dual of [(257, 4), 1019, 10]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1019,256) [i]
- linear OOA(2564, 257, F256, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25613, 514, F256, 4, 9) (dual of [(514, 4), 2043, 10]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12815, 514, S128, 4, 9), using
(15−9, 15, 413506)-Net in Base 128 — Upper bound on s
There is no (6, 15, 413507)-net in base 128, because
- 1 times m-reduction [i] would yield (6, 14, 413507)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 316914 083492 743730 147050 651152 > 12814 [i]