Best Known (16, 23, s)-Nets in Base 4
(16, 23, 195)-Net over F4 — Constructive and digital
Digital (16, 23, 195)-net over F4, using
- 1 times m-reduction [i] based on digital (16, 24, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 8, 65)-net over F64, using
(16, 23, 273)-Net over F4 — Digital
Digital (16, 23, 273)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(423, 273, F4, 7) (dual of [273, 250, 8]-code), using
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(422, 266, F4, 7) (dual of [266, 244, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(421, 256, F4, 7) (dual of [256, 235, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(413, 256, F4, 5) (dual of [256, 243, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(49, 10, F4, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,4)), using
- dual of repetition code with length 10 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(422, 266, F4, 7) (dual of [266, 244, 8]-code), using
(16, 23, 15751)-Net in Base 4 — Upper bound on s
There is no (16, 23, 15752)-net in base 4, because
- 1 times m-reduction [i] would yield (16, 22, 15752)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 17 594828 110333 > 422 [i]