Best Known (126, 161, s)-Nets in Base 4
(126, 161, 1048)-Net over F4 — Constructive and digital
Digital (126, 161, 1048)-net over F4, using
- 41 times duplication [i] based on digital (125, 160, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 40, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 40, 262)-net over F256, using
(126, 161, 3616)-Net over F4 — Digital
Digital (126, 161, 3616)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4161, 3616, F4, 35) (dual of [3616, 3455, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 4114, F4, 35) (dual of [4114, 3953, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4145, 4096, F4, 33) (dual of [4096, 3951, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4139, 4096, F4, 31) (dual of [4096, 3957, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(34) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4161, 4114, F4, 35) (dual of [4114, 3953, 36]-code), using
(126, 161, 1109853)-Net in Base 4 — Upper bound on s
There is no (126, 161, 1109854)-net in base 4, because
- 1 times m-reduction [i] would yield (126, 160, 1109854)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 136009 624946 449783 731757 587455 547078 547280 174365 617034 850292 455903 512849 290137 996333 671124 752085 > 4160 [i]