Best Known (49, 49+22, s)-Nets in Base 4
(49, 49+22, 240)-Net over F4 — Constructive and digital
Digital (49, 71, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (49, 72, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 24, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 24, 80)-net over F64, using
(49, 49+22, 325)-Net over F4 — Digital
Digital (49, 71, 325)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(471, 325, F4, 22) (dual of [325, 254, 23]-code), using
- 57 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 14 times 0, 1, 16 times 0) [i] based on linear OA(463, 260, F4, 22) (dual of [260, 197, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(463, 256, F4, 22) (dual of [256, 193, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 4, F4, 0) (dual of [4, 4, 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
- 57 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 14 times 0, 1, 16 times 0) [i] based on linear OA(463, 260, F4, 22) (dual of [260, 197, 23]-code), using
(49, 49+22, 12578)-Net in Base 4 — Upper bound on s
There is no (49, 71, 12579)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5 579236 470292 585885 868639 962996 378836 096688 > 471 [i]