Best Known (92−15, 92, s)-Nets in Base 3
(92−15, 92, 2814)-Net over F3 — Constructive and digital
Digital (77, 92, 2814)-net over F3, using
- net defined by OOA [i] based on linear OOA(392, 2814, F3, 15, 15) (dual of [(2814, 15), 42118, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(392, 19699, F3, 15) (dual of [19699, 19607, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(392, 19703, F3, 15) (dual of [19703, 19611, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(391, 19684, F3, 15) (dual of [19684, 19593, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(392, 19703, F3, 15) (dual of [19703, 19611, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(392, 19699, F3, 15) (dual of [19699, 19607, 16]-code), using
(92−15, 92, 9852)-Net over F3 — Digital
Digital (77, 92, 9852)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(392, 9852, F3, 2, 15) (dual of [(9852, 2), 19612, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(392, 19704, F3, 15) (dual of [19704, 19612, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(391, 19684, F3, 15) (dual of [19684, 19593, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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 X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(392, 19704, F3, 15) (dual of [19704, 19612, 16]-code), using
(92−15, 92, 2694411)-Net in Base 3 — Upper bound on s
There is no (77, 92, 2694412)-net in base 3, because
- 1 times m-reduction [i] would yield (77, 91, 2694412)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 26 183927 826070 411398 229029 487899 450199 988817 > 391 [i]