Best Known (172, 210, s)-Nets in Base 4
(172, 210, 1539)-Net over F4 — Constructive and digital
Digital (172, 210, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (172, 216, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 72, 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, 72, 513)-net over F64, using
(172, 210, 14862)-Net over F4 — Digital
Digital (172, 210, 14862)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4210, 14862, F4, 38) (dual of [14862, 14652, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4210, 16432, F4, 38) (dual of [16432, 16222, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- 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(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(413, 48, F4, 6) (dual of [48, 35, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4210, 16432, F4, 38) (dual of [16432, 16222, 39]-code), using
(172, 210, large)-Net in Base 4 — Upper bound on s
There is no (172, 210, large)-net in base 4, because
- 36 times m-reduction [i] would yield (172, 174, large)-net in base 4, but