Best Known (108−22, 108, s)-Nets in Base 4
(108−22, 108, 1051)-Net over F4 — Constructive and digital
Digital (86, 108, 1051)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 20, 23)-net over F4, using
- 3 times m-reduction [i] based on digital (9, 23, 23)-net over F4, using
- digital (66, 88, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 257)-net over F256, using
- digital (9, 20, 23)-net over F4, using
(108−22, 108, 4249)-Net over F4 — Digital
Digital (86, 108, 4249)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4108, 4249, F4, 22) (dual of [4249, 4141, 23]-code), using
- 136 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0) [i] based on linear OA(497, 4102, F4, 22) (dual of [4102, 4005, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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(21) ⊂ Ce(20) [i] based on
- 136 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0) [i] based on linear OA(497, 4102, F4, 22) (dual of [4102, 4005, 23]-code), using
(108−22, 108, 1333596)-Net in Base 4 — Upper bound on s
There is no (86, 108, 1333597)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 105312 479493 567734 735232 068482 027978 403786 309300 488367 701223 983252 > 4108 [i]