Best Known (190−40, 190, s)-Nets in Base 4
(190−40, 190, 1056)-Net over F4 — Constructive and digital
Digital (150, 190, 1056)-net over F4, using
- 42 times duplication [i] based on digital (148, 188, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 47, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 47, 264)-net over F256, using
(190−40, 190, 4443)-Net over F4 — Digital
Digital (150, 190, 4443)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4190, 4443, F4, 40) (dual of [4443, 4253, 41]-code), using
- 326 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 32 times 0, 1, 56 times 0, 1, 88 times 0, 1, 119 times 0) [i] based on linear OA(4182, 4109, F4, 40) (dual of [4109, 3927, 41]-code), using
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- 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(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- 326 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 32 times 0, 1, 56 times 0, 1, 88 times 0, 1, 119 times 0) [i] based on linear OA(4182, 4109, F4, 40) (dual of [4109, 3927, 41]-code), using
(190−40, 190, 1451276)-Net in Base 4 — Upper bound on s
There is no (150, 190, 1451277)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2 462654 494128 519304 881115 405492 458243 489575 463914 438266 086228 144919 318637 998408 095107 327190 474628 811839 227699 413296 > 4190 [i]