Best Known (163, 163+37, s)-Nets in Base 4
(163, 163+37, 1539)-Net over F4 — Constructive and digital
Digital (163, 200, 1539)-net over F4, using
- t-expansion [i] based on digital (162, 200, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (162, 201, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 67, 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, 67, 513)-net over F64, using
- 1 times m-reduction [i] based on digital (162, 201, 1539)-net over F4, using
(163, 163+37, 12254)-Net over F4 — Digital
Digital (163, 200, 12254)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4200, 12254, F4, 37) (dual of [12254, 12054, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4200, 16422, F4, 37) (dual of [16422, 16222, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- 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(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(410, 38, F4, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4200, 16422, F4, 37) (dual of [16422, 16222, 38]-code), using
(163, 163+37, large)-Net in Base 4 — Upper bound on s
There is no (163, 200, large)-net in base 4, because
- 35 times m-reduction [i] would yield (163, 165, large)-net in base 4, but