Best Known (37, 37+112, s)-Nets in Base 4
(37, 37+112, 56)-Net over F4 — Constructive and digital
Digital (37, 149, 56)-net over F4, using
- t-expansion [i] based on digital (33, 149, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(37, 37+112, 66)-Net over F4 — Digital
Digital (37, 149, 66)-net over F4, using
- net from sequence [i] based on digital (37, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 37 and N(F) ≥ 66, using
(37, 37+112, 156)-Net over F4 — Upper bound on s (digital)
There is no digital (37, 149, 157)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4149, 157, F4, 112) (dual of [157, 8, 113]-code), but
- construction Y1 [i] would yield
- linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- residual code [i] would yield linear OA(48, 11, F4, 7) (dual of [11, 3, 8]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- OA(48, 157, S4, 4), but
- discarding factors would yield OA(48, 121, S4, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 65704 > 48 [i]
- discarding factors would yield OA(48, 121, S4, 4), but
- linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- construction Y1 [i] would yield
(37, 37+112, 245)-Net in Base 4 — Upper bound on s
There is no (37, 149, 246)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 530504 814532 232289 457275 279277 453467 593524 736705 228741 409647 765876 815367 497379 292462 103864 > 4149 [i]