Best Known (93, 93+16, s)-Nets in Base 3
(93, 93+16, 7385)-Net over F3 — Constructive and digital
Digital (93, 109, 7385)-net over F3, using
- net defined by OOA [i] based on linear OOA(3109, 7385, F3, 16, 16) (dual of [(7385, 16), 118051, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3109, 59080, F3, 16) (dual of [59080, 58971, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3109, 59087, F3, 16) (dual of [59087, 58978, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(371, 59049, F3, 11) (dual of [59049, 58978, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3109, 59087, F3, 16) (dual of [59087, 58978, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3109, 59080, F3, 16) (dual of [59080, 58971, 17]-code), using
(93, 93+16, 23946)-Net over F3 — Digital
Digital (93, 109, 23946)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3109, 23946, F3, 2, 16) (dual of [(23946, 2), 47783, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3109, 29543, F3, 2, 16) (dual of [(29543, 2), 58977, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3109, 59086, F3, 16) (dual of [59086, 58977, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3109, 59087, F3, 16) (dual of [59087, 58978, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(371, 59049, F3, 11) (dual of [59049, 58978, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3109, 59087, F3, 16) (dual of [59087, 58978, 17]-code), using
- OOA 2-folding [i] based on linear OA(3109, 59086, F3, 16) (dual of [59086, 58977, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(3109, 29543, F3, 2, 16) (dual of [(29543, 2), 58977, 17]-NRT-code), using
(93, 93+16, 5962612)-Net in Base 3 — Upper bound on s
There is no (93, 109, 5962613)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 10144 176408 562779 349566 374385 894766 129896 593633 888097 > 3109 [i]