Best Known (106, 106+29, s)-Nets in Base 4
(106, 106+29, 1045)-Net over F4 — Constructive and digital
Digital (106, 135, 1045)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (87, 116, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 29, 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
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- digital (5, 19, 17)-net over F4, using
(106, 106+29, 3521)-Net over F4 — Digital
Digital (106, 135, 3521)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4135, 3521, F4, 29) (dual of [3521, 3386, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4135, 4124, F4, 29) (dual of [4124, 3989, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(45, 25, F4, 3) (dual of [25, 20, 4]-code or 25-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4135, 4124, F4, 29) (dual of [4124, 3989, 30]-code), using
(106, 106+29, 1166558)-Net in Base 4 — Upper bound on s
There is no (106, 135, 1166559)-net in base 4, because
- 1 times m-reduction [i] would yield (106, 134, 1166559)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 474 287124 297340 114017 749054 194780 855367 716630 155221 296500 137352 072675 195948 029395 > 4134 [i]