Best Known (187, 187+44, s)-Nets in Base 3
(187, 187+44, 896)-Net over F3 — Constructive and digital
Digital (187, 231, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (187, 232, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 58, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 58, 224)-net over F81, using
(187, 187+44, 3296)-Net over F3 — Digital
Digital (187, 231, 3296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3231, 3296, F3, 44) (dual of [3296, 3065, 45]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(3230, 3291, F3, 44) (dual of [3291, 3061, 45]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(3229, 3289, F3, 44) (dual of [3289, 3060, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3229, 3281, F3, 44) (dual of [3281, 3052, 45]-code), using an extension Ce(43) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3221, 3281, F3, 43) (dual of [3281, 3060, 44]-code), using an extension Ce(42) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3229, 3290, F3, 43) (dual of [3290, 3061, 44]-code), using Gilbert–Varšamov bound and bm = 3229 > Vbs−1(k−1) = 12 615641 539872 118256 399378 807082 663458 339630 606048 599909 154946 097516 291712 410334 781485 058789 550974 366987 878307 [i]
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s (see above)
- linear OA(3229, 3289, F3, 44) (dual of [3289, 3060, 45]-code), using
- construction X with Varšamov bound [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(3230, 3291, F3, 44) (dual of [3291, 3061, 45]-code), using
(187, 187+44, 462994)-Net in Base 3 — Upper bound on s
There is no (187, 231, 462995)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 164 064868 310175 452588 202381 867485 587360 887693 744194 816373 661858 672764 976705 371382 641171 824400 620456 910905 963933 > 3231 [i]