Best Known (13, 31, s)-Nets in Base 128
(13, 31, 342)-Net over F128 — Constructive and digital
Digital (13, 31, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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 (3, 21, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (1, 10, 150)-net over F128, using
(13, 31, 411)-Net over F128 — Digital
Digital (13, 31, 411)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12831, 411, F128, 18) (dual of [411, 380, 19]-code), using
- 25 step Varšamov–Edel lengthening with (ri) = (1, 24 times 0) [i] based on linear OA(12830, 385, F128, 18) (dual of [385, 355, 19]-code), using
- construction XX applied to C1 = C([380,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([380,16]) [i] based on
- linear OA(12828, 381, F128, 17) (dual of [381, 353, 18]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12828, 381, F128, 17) (dual of [381, 353, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12830, 381, F128, 18) (dual of [381, 351, 19]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12826, 381, F128, 16) (dual of [381, 355, 17]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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([380,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([380,16]) [i] based on
- 25 step Varšamov–Edel lengthening with (ri) = (1, 24 times 0) [i] based on linear OA(12830, 385, F128, 18) (dual of [385, 355, 19]-code), using
(13, 31, 514)-Net in Base 128 — Constructive
(13, 31, 514)-net in base 128, using
- 1 times m-reduction [i] based on (13, 32, 514)-net in base 128, using
- base change [i] based on digital (9, 28, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (0, 19, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 9, 257)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (9, 28, 514)-net over F256, using
(13, 31, 591714)-Net in Base 128 — Upper bound on s
There is no (13, 31, 591715)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 210625 309036 785841 603665 425625 465454 081272 080897 228757 951192 273432 > 12831 [i]