Best Known (219−36, 219, s)-Nets in Base 3
(219−36, 219, 1480)-Net over F3 — Constructive and digital
Digital (183, 219, 1480)-net over F3, using
- t-expansion [i] based on digital (181, 219, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (181, 220, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F81, using
- 1 times m-reduction [i] based on digital (181, 220, 1480)-net over F3, using
(219−36, 219, 9000)-Net over F3 — Digital
Digital (183, 219, 9000)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3219, 9000, F3, 2, 36) (dual of [(9000, 2), 17781, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3219, 9851, F3, 2, 36) (dual of [(9851, 2), 19483, 37]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3218, 9851, F3, 2, 36) (dual of [(9851, 2), 19484, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3218, 19702, F3, 36) (dual of [19702, 19484, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(3218, 19702, F3, 36) (dual of [19702, 19484, 37]-code), using
- 31 times duplication [i] based on linear OOA(3218, 9851, F3, 2, 36) (dual of [(9851, 2), 19484, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3219, 9851, F3, 2, 36) (dual of [(9851, 2), 19483, 37]-NRT-code), using
(219−36, 219, 2410305)-Net in Base 3 — Upper bound on s
There is no (183, 219, 2410306)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 308 713009 995193 941628 153544 953566 239438 267317 381684 767014 109618 014097 966025 567254 694496 945952 018023 478837 > 3219 [i]