Best Known (147, 147+38, s)-Nets in Base 4
(147, 147+38, 1060)-Net over F4 — Constructive and digital
Digital (147, 185, 1060)-net over F4, using
- 41 times duplication [i] based on digital (146, 184, 1060)-net over F4, using
- trace code for nets [i] based on digital (8, 46, 265)-net over F256, using
- net from sequence [i] based on digital (8, 264)-sequence over F256, using
- trace code for nets [i] based on digital (8, 46, 265)-net over F256, using
(147, 147+38, 5023)-Net over F4 — Digital
Digital (147, 185, 5023)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4185, 5023, F4, 38) (dual of [5023, 4838, 39]-code), using
- 905 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 31 times 0, 1, 47 times 0, 1, 68 times 0, 1, 95 times 0, 1, 121 times 0, 1, 145 times 0, 1, 164 times 0, 1, 177 times 0) [i] based on linear OA(4169, 4102, F4, 38) (dual of [4102, 3933, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(4169, 4096, F4, 38) (dual of [4096, 3927, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(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(37) ⊂ Ce(36) [i] based on
- 905 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 31 times 0, 1, 47 times 0, 1, 68 times 0, 1, 95 times 0, 1, 121 times 0, 1, 145 times 0, 1, 164 times 0, 1, 177 times 0) [i] based on linear OA(4169, 4102, F4, 38) (dual of [4102, 3933, 39]-code), using
(147, 147+38, 1924216)-Net in Base 4 — Upper bound on s
There is no (147, 185, 1924217)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2404 926837 661876 848320 261141 104759 624932 306789 415114 582911 292349 264255 962692 560340 315959 622486 655971 763686 541048 > 4185 [i]