Best Known (188, 188+43, s)-Nets in Base 4
(188, 188+43, 1539)-Net over F4 — Constructive and digital
Digital (188, 231, 1539)-net over F4, using
- 9 times m-reduction [i] based on digital (188, 240, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 80, 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, 80, 513)-net over F64, using
(188, 188+43, 12796)-Net over F4 — Digital
Digital (188, 231, 12796)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4231, 12796, F4, 43) (dual of [12796, 12565, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 16412, F4, 43) (dual of [16412, 16181, 44]-code), using
- construction XX applied to Ce(42) ⊂ Ce(38) ⊂ Ce(37) [i] based on
- linear OA(4225, 16384, F4, 43) (dual of [16384, 16159, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4204, 16384, F4, 39) (dual of [16384, 16180, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4197, 16384, F4, 38) (dual of [16384, 16187, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), 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(42) ⊂ Ce(38) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(4231, 16412, F4, 43) (dual of [16412, 16181, 44]-code), using
(188, 188+43, large)-Net in Base 4 — Upper bound on s
There is no (188, 231, large)-net in base 4, because
- 41 times m-reduction [i] would yield (188, 190, large)-net in base 4, but