Best Known (77, 93, s)-Nets in Base 3
(77, 93, 2461)-Net over F3 — Constructive and digital
Digital (77, 93, 2461)-net over F3, using
- 31 times duplication [i] based on digital (76, 92, 2461)-net over F3, using
- net defined by OOA [i] based on linear OOA(392, 2461, F3, 16, 16) (dual of [(2461, 16), 39284, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(392, 19688, F3, 16) (dual of [19688, 19596, 17]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(392, 19693, F3, 16) (dual of [19693, 19601, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(392, 19688, F3, 16) (dual of [19688, 19596, 17]-code), using
- net defined by OOA [i] based on linear OOA(392, 2461, F3, 16, 16) (dual of [(2461, 16), 39284, 17]-NRT-code), using
(77, 93, 6564)-Net over F3 — Digital
Digital (77, 93, 6564)-net over F3, using
- 31 times duplication [i] based on digital (76, 92, 6564)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(392, 6564, F3, 3, 16) (dual of [(6564, 3), 19600, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(392, 19692, F3, 16) (dual of [19692, 19600, 17]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(392, 19693, F3, 16) (dual of [19693, 19601, 17]-code), using
- OOA 3-folding [i] based on linear OA(392, 19692, F3, 16) (dual of [19692, 19600, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(392, 6564, F3, 3, 16) (dual of [(6564, 3), 19600, 17]-NRT-code), using
(77, 93, 662505)-Net in Base 3 — Upper bound on s
There is no (77, 93, 662506)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 235 655347 932693 886969 412164 271871 317129 468625 > 393 [i]