Best Known (161, 161+38, s)-Nets in Base 4
(161, 161+38, 1539)-Net over F4 — Constructive and digital
Digital (161, 199, 1539)-net over F4, using
- 41 times duplication [i] based on digital (160, 198, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 66, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 66, 513)-net over F64, using
(161, 161+38, 9720)-Net over F4 — Digital
Digital (161, 199, 9720)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4199, 9720, F4, 38) (dual of [9720, 9521, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, 16394, F4, 38) (dual of [16394, 16195, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4197, 16384, F4, 38) (dual of [16384, 16187, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4183, 16384, F4, 35) (dual of [16384, 16201, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4199, 16394, F4, 38) (dual of [16394, 16195, 39]-code), using
(161, 161+38, 5344157)-Net in Base 4 — Upper bound on s
There is no (161, 199, 5344158)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 645564 136250 297890 835415 238334 013008 441476 080866 801617 277875 820285 894987 932757 829035 659457 780503 814977 067648 241423 079798 > 4199 [i]