Best Known (79, 108, s)-Nets in Base 4
(79, 108, 531)-Net over F4 — Constructive and digital
Digital (79, 108, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 36, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
(79, 108, 865)-Net over F4 — Digital
Digital (79, 108, 865)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4108, 865, F4, 29) (dual of [865, 757, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, 1032, F4, 29) (dual of [1032, 924, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4101, 1024, F4, 27) (dual of [1024, 923, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 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(41, 7, F4, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4108, 1032, F4, 29) (dual of [1032, 924, 30]-code), using
(79, 108, 80488)-Net in Base 4 — Upper bound on s
There is no (79, 108, 80489)-net in base 4, because
- 1 times m-reduction [i] would yield (79, 107, 80489)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 26329 589116 444170 039873 296170 412769 940141 834958 369848 206929 644432 > 4107 [i]