Best Known (10, 17, s)-Nets in Base 4
(10, 17, 48)-Net over F4 — Constructive and digital
Digital (10, 17, 48)-net over F4, using
- 1 times m-reduction [i] based on digital (10, 18, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 9, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 9, 24)-net over F16, using
(10, 17, 71)-Net over F4 — Digital
Digital (10, 17, 71)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(417, 71, F4, 7) (dual of [71, 54, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
(10, 17, 982)-Net in Base 4 — Upper bound on s
There is no (10, 17, 983)-net in base 4, because
- 1 times m-reduction [i] would yield (10, 16, 983)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4300 500160 > 416 [i]