Best Known (235−45, 235, s)-Nets in Base 4
(235−45, 235, 1539)-Net over F4 — Constructive and digital
Digital (190, 235, 1539)-net over F4, using
- 8 times m-reduction [i] based on digital (190, 243, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 81, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 81, 513)-net over F64, using
(235−45, 235, 10599)-Net over F4 — Digital
Digital (190, 235, 10599)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4235, 10599, F4, 45) (dual of [10599, 10364, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4235, 16401, F4, 45) (dual of [16401, 16166, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(41) [i] based on
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(44) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(4235, 16401, F4, 45) (dual of [16401, 16166, 46]-code), using
(235−45, 235, 7646448)-Net in Base 4 — Upper bound on s
There is no (190, 235, 7646449)-net in base 4, because
- 1 times m-reduction [i] would yield (190, 234, 7646449)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 762 147118 124795 016594 753666 874938 517843 864088 465836 652322 850297 637248 180320 624346 819870 228113 342937 644279 292458 531547 047551 930452 960851 850960 > 4234 [i]