Best Known (155, 155+37, s)-Nets in Base 4
(155, 155+37, 1118)-Net over F4 — Constructive and digital
Digital (155, 192, 1118)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (26, 44, 90)-net over F4, using
- trace code for nets [i] based on digital (4, 22, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- trace code for nets [i] based on digital (4, 22, 45)-net over F16, using
- digital (111, 148, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 37, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 37, 257)-net over F256, using
- digital (26, 44, 90)-net over F4, using
(155, 155+37, 8919)-Net over F4 — Digital
Digital (155, 192, 8919)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4192, 8919, F4, 37) (dual of [8919, 8727, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4192, 16394, F4, 37) (dual of [16394, 16202, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- 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(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(36) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4192, 16394, F4, 37) (dual of [16394, 16202, 38]-code), using
(155, 155+37, 6159326)-Net in Base 4 — Upper bound on s
There is no (155, 192, 6159327)-net in base 4, because
- 1 times m-reduction [i] would yield (155, 191, 6159327)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 9 850509 055809 658863 746294 398096 837947 733556 181530 368177 830843 919399 895434 990494 350718 895143 618646 222434 489132 981877 > 4191 [i]