Best Known (88, 104, s)-Nets in Base 3
(88, 104, 7382)-Net over F3 — Constructive and digital
Digital (88, 104, 7382)-net over F3, using
- 32 times duplication [i] based on digital (86, 102, 7382)-net over F3, using
- net defined by OOA [i] based on linear OOA(3102, 7382, F3, 16, 16) (dual of [(7382, 16), 118010, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3102, 59056, F3, 16) (dual of [59056, 58954, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3102, 59056, F3, 16) (dual of [59056, 58954, 17]-code), using
- net defined by OOA [i] based on linear OOA(3102, 7382, F3, 16, 16) (dual of [(7382, 16), 118010, 17]-NRT-code), using
(88, 104, 19687)-Net over F3 — Digital
Digital (88, 104, 19687)-net over F3, using
- 31 times duplication [i] based on digital (87, 103, 19687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3103, 19687, F3, 3, 16) (dual of [(19687, 3), 58958, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3103, 59061, F3, 16) (dual of [59061, 58958, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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 code embedding in larger space [i] based on linear OA(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- OOA 3-folding [i] based on linear OA(3103, 59061, F3, 16) (dual of [59061, 58958, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3103, 19687, F3, 3, 16) (dual of [(19687, 3), 58958, 17]-NRT-code), using
(88, 104, 3000787)-Net in Base 3 — Upper bound on s
There is no (88, 104, 3000788)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 41 745593 598994 266003 870561 838685 898960 877381 601217 > 3104 [i]