Best Known (218−41, 218, s)-Nets in Base 4
(218−41, 218, 1539)-Net over F4 — Constructive and digital
Digital (177, 218, 1539)-net over F4, using
- t-expansion [i] based on digital (176, 218, 1539)-net over F4, using
- 4 times m-reduction [i] based on digital (176, 222, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 74, 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, 74, 513)-net over F64, using
- 4 times m-reduction [i] based on digital (176, 222, 1539)-net over F4, using
(218−41, 218, 11456)-Net over F4 — Digital
Digital (177, 218, 11456)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4218, 11456, F4, 41) (dual of [11456, 11238, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 16413, F4, 41) (dual of [16413, 16195, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4183, 16384, F4, 35) (dual of [16384, 16201, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 16413, F4, 41) (dual of [16413, 16195, 42]-code), using
(218−41, 218, large)-Net in Base 4 — Upper bound on s
There is no (177, 218, large)-net in base 4, because
- 39 times m-reduction [i] would yield (177, 179, large)-net in base 4, but