Best Known (61−18, 61, s)-Nets in Base 4
(61−18, 61, 240)-Net over F4 — Constructive and digital
Digital (43, 61, 240)-net over F4, using
- 2 times m-reduction [i] based on digital (43, 63, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 21, 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, 21, 80)-net over F64, using
(61−18, 61, 356)-Net over F4 — Digital
Digital (43, 61, 356)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(461, 356, F4, 18) (dual of [356, 295, 19]-code), using
- 88 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 19 times 0, 1, 23 times 0) [i] based on linear OA(451, 258, F4, 18) (dual of [258, 207, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(451, 256, F4, 18) (dual of [256, 205, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(449, 256, F4, 17) (dual of [256, 207, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(17) ⊂ Ce(16) [i] based on
- 88 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 19 times 0, 1, 23 times 0) [i] based on linear OA(451, 258, F4, 18) (dual of [258, 207, 19]-code), using
(61−18, 61, 16637)-Net in Base 4 — Upper bound on s
There is no (43, 61, 16638)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5 319639 257519 530883 781783 021890 051695 > 461 [i]