Best Known (135, 135+37, s)-Nets in Base 4
(135, 135+37, 1052)-Net over F4 — Constructive and digital
Digital (135, 172, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 43, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(135, 135+37, 4024)-Net over F4 — Digital
Digital (135, 172, 4024)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4172, 4024, F4, 37) (dual of [4024, 3852, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 4126, F4, 37) (dual of [4126, 3954, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- construction XX applied to Ce(36) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 4126, F4, 37) (dual of [4126, 3954, 38]-code), using
(135, 135+37, 1320000)-Net in Base 4 — Upper bound on s
There is no (135, 172, 1320001)-net in base 4, because
- 1 times m-reduction [i] would yield (135, 171, 1320001)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 8 959094 541821 967963 237783 042956 310800 422368 838570 174957 054916 819774 501124 844806 570896 522254 293561 960736 > 4171 [i]