Best Known (80, 80+26, s)-Nets in Base 4
(80, 80+26, 1028)-Net over F4 — Constructive and digital
Digital (80, 106, 1028)-net over F4, using
- 42 times duplication [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
(80, 80+26, 1227)-Net over F4 — Digital
Digital (80, 106, 1227)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4106, 1227, F4, 26) (dual of [1227, 1121, 27]-code), using
- 188 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 34 times 0, 1, 48 times 0, 1, 57 times 0) [i] based on linear OA(496, 1029, F4, 26) (dual of [1029, 933, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(496, 1024, F4, 26) (dual of [1024, 928, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(491, 1024, F4, 25) (dual of [1024, 933, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 188 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 34 times 0, 1, 48 times 0, 1, 57 times 0) [i] based on linear OA(496, 1029, F4, 26) (dual of [1029, 933, 27]-code), using
(80, 80+26, 153235)-Net in Base 4 — Upper bound on s
There is no (80, 106, 153236)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6582 141016 929921 999945 261809 489100 833735 292444 359178 999559 869412 > 4106 [i]