Best Known (76, 76+15, s)-Nets in Base 3
(76, 76+15, 2813)-Net over F3 — Constructive and digital
Digital (76, 91, 2813)-net over F3, using
- net defined by OOA [i] based on linear OOA(391, 2813, F3, 15, 15) (dual of [(2813, 15), 42104, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(391, 19692, F3, 15) (dual of [19692, 19601, 16]-code), using
- 1 times truncation [i] based on linear OA(392, 19693, F3, 16) (dual of [19693, 19601, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(392, 19693, F3, 16) (dual of [19693, 19601, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(391, 19692, F3, 15) (dual of [19692, 19601, 16]-code), using
(76, 76+15, 9130)-Net over F3 — Digital
Digital (76, 91, 9130)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(391, 9130, F3, 2, 15) (dual of [(9130, 2), 18169, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(391, 9846, F3, 2, 15) (dual of [(9846, 2), 19601, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(391, 19692, F3, 15) (dual of [19692, 19601, 16]-code), using
- 1 times truncation [i] based on linear OA(392, 19693, F3, 16) (dual of [19693, 19601, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(392, 19693, F3, 16) (dual of [19693, 19601, 17]-code), using
- OOA 2-folding [i] based on linear OA(391, 19692, F3, 15) (dual of [19692, 19601, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(391, 9846, F3, 2, 15) (dual of [(9846, 2), 19601, 16]-NRT-code), using
(76, 76+15, 2303051)-Net in Base 3 — Upper bound on s
There is no (76, 91, 2303052)-net in base 3, because
- 1 times m-reduction [i] would yield (76, 90, 2303052)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8 727989 608987 743166 765897 193891 364248 140113 > 390 [i]