Best Known (238−39, 238, s)-Nets in Base 3
(238−39, 238, 1480)-Net over F3 — Constructive and digital
Digital (199, 238, 1480)-net over F3, using
- 6 times m-reduction [i] based on digital (199, 244, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 61, 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, 61, 370)-net over F81, using
(238−39, 238, 9548)-Net over F3 — Digital
Digital (199, 238, 9548)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3238, 9548, F3, 2, 39) (dual of [(9548, 2), 18858, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3238, 9852, F3, 2, 39) (dual of [(9852, 2), 19466, 40]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3237, 9852, F3, 2, 39) (dual of [(9852, 2), 19467, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3237, 19704, F3, 39) (dual of [19704, 19467, 40]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3236, 19703, F3, 39) (dual of [19703, 19467, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(3235, 19684, F3, 39) (dual of [19684, 19449, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3217, 19684, F3, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [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 C([0,19]) ⊂ C([0,18]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3236, 19703, F3, 39) (dual of [19703, 19467, 40]-code), using
- OOA 2-folding [i] based on linear OA(3237, 19704, F3, 39) (dual of [19704, 19467, 40]-code), using
- 31 times duplication [i] based on linear OOA(3237, 9852, F3, 2, 39) (dual of [(9852, 2), 19467, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3238, 9852, F3, 2, 39) (dual of [(9852, 2), 19466, 40]-NRT-code), using
(238−39, 238, 3545219)-Net in Base 3 — Upper bound on s
There is no (199, 238, 3545220)-net in base 3, because
- 1 times m-reduction [i] would yield (199, 237, 3545220)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 119602 147635 196920 236728 873655 594157 345688 406522 703807 625584 114263 322133 485897 762731 158088 141596 199914 696650 717105 > 3237 [i]