Best Known (11, 11+14, s)-Nets in Base 64
(11, 11+14, 195)-Net over F64 — Constructive and digital
Digital (11, 25, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (0, 7, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 14, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 4, 65)-net over F64, using
(11, 11+14, 261)-Net in Base 64 — Constructive
(11, 25, 261)-net in base 64, using
- 3 times m-reduction [i] based on (11, 28, 261)-net in base 64, using
- base change [i] based on digital (4, 21, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- base change [i] based on digital (4, 21, 261)-net over F256, using
(11, 11+14, 319)-Net over F64 — Digital
Digital (11, 25, 319)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6425, 319, F64, 14) (dual of [319, 294, 15]-code), using
- construction XX applied to C1 = C([314,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([314,12]) [i] based on
- linear OA(6423, 315, F64, 13) (dual of [315, 292, 14]-code), using the BCH-code C(I) with length 315 | 642−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6423, 315, F64, 13) (dual of [315, 292, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 315 | 642−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6425, 315, F64, 14) (dual of [315, 290, 15]-code), using the BCH-code C(I) with length 315 | 642−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(6421, 315, F64, 12) (dual of [315, 294, 13]-code), using the expurgated narrow-sense BCH-code C(I) with length 315 | 642−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([314,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([314,12]) [i] based on
(11, 11+14, 321)-Net in Base 64
(11, 25, 321)-net in base 64, using
- 11 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
- base change [i] based on digital (2, 27, 321)-net over F256, using
- net from sequence [i] based on digital (2, 320)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- net from sequence [i] based on digital (2, 320)-sequence over F256, using
- base change [i] based on digital (2, 27, 321)-net over F256, using
(11, 11+14, 151430)-Net in Base 64 — Upper bound on s
There is no (11, 25, 151431)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1427 313252 238008 716094 211145 733296 198340 642112 > 6425 [i]