Best Known (65, 65+22, s)-Nets in Base 4
(65, 65+22, 531)-Net over F4 — Constructive and digital
Digital (65, 87, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 29, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
(65, 65+22, 1056)-Net over F4 — Digital
Digital (65, 87, 1056)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(487, 1056, F4, 22) (dual of [1056, 969, 23]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0) [i] based on linear OA(481, 1029, F4, 22) (dual of [1029, 948, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(481, 1024, F4, 22) (dual of [1024, 943, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 21 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0) [i] based on linear OA(481, 1029, F4, 22) (dual of [1029, 948, 23]-code), using
(65, 65+22, 94536)-Net in Base 4 — Upper bound on s
There is no (65, 87, 94537)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 23946 165616 442646 257518 597666 132220 327890 953910 606336 > 487 [i]