Best Known (23, 23+10, s)-Nets in Base 4
(23, 23+10, 240)-Net over F4 — Constructive and digital
Digital (23, 33, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 11, 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
(23, 23+10, 280)-Net over F4 — Digital
Digital (23, 33, 280)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(433, 280, F4, 10) (dual of [280, 247, 11]-code), using
- 16 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 8 times 0) [i] based on linear OA(429, 260, F4, 10) (dual of [260, 231, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(429, 256, F4, 10) (dual of [256, 227, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(425, 256, F4, 9) (dual of [256, 231, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- 16 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 8 times 0) [i] based on linear OA(429, 260, F4, 10) (dual of [260, 231, 11]-code), using
(23, 23+10, 8168)-Net in Base 4 — Upper bound on s
There is no (23, 33, 8169)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 73 831855 078274 547808 > 433 [i]