Best Known (248−45, 248, s)-Nets in Base 4
(248−45, 248, 1560)-Net over F4 — Constructive and digital
Digital (203, 248, 1560)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 29, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (174, 219, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 73, 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, 73, 513)-net over F64, using
- digital (7, 29, 21)-net over F4, using
(248−45, 248, 16135)-Net over F4 — Digital
Digital (203, 248, 16135)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4248, 16135, F4, 45) (dual of [16135, 15887, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4248, 16442, F4, 45) (dual of [16442, 16194, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(36) [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(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(416, 58, F4, 7) (dual of [58, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using
- construction X applied to Ce(44) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(4248, 16442, F4, 45) (dual of [16442, 16194, 46]-code), using
(248−45, 248, large)-Net in Base 4 — Upper bound on s
There is no (203, 248, large)-net in base 4, because
- 43 times m-reduction [i] would yield (203, 205, large)-net in base 4, but