Best Known (78, 78+25, s)-Nets in Base 4
(78, 78+25, 1028)-Net over F4 — Constructive and digital
Digital (78, 103, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (78, 104, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 26, 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, 26, 257)-net over F256, using
(78, 78+25, 1272)-Net over F4 — Digital
Digital (78, 103, 1272)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4103, 1272, F4, 25) (dual of [1272, 1169, 26]-code), using
- 235 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 29 times 0, 1, 40 times 0, 1, 51 times 0, 1, 61 times 0) [i] based on linear OA(491, 1025, F4, 25) (dual of [1025, 934, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 235 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 29 times 0, 1, 40 times 0, 1, 51 times 0, 1, 61 times 0) [i] based on linear OA(491, 1025, F4, 25) (dual of [1025, 934, 26]-code), using
(78, 78+25, 231063)-Net in Base 4 — Upper bound on s
There is no (78, 103, 231064)-net in base 4, because
- 1 times m-reduction [i] would yield (78, 102, 231064)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 25 711049 196623 654915 073196 143102 009980 035181 699186 491929 756670 > 4102 [i]