Best Known (111−17, 111, s)-Nets in Base 3
(111−17, 111, 7382)-Net over F3 — Constructive and digital
Digital (94, 111, 7382)-net over F3, using
- net defined by OOA [i] based on linear OOA(3111, 7382, F3, 17, 17) (dual of [(7382, 17), 125383, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3111, 59057, F3, 17) (dual of [59057, 58946, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, 59059, F3, 17) (dual of [59059, 58948, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3111, 59059, F3, 17) (dual of [59059, 58948, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3111, 59057, F3, 17) (dual of [59057, 58946, 18]-code), using
(111−17, 111, 19686)-Net over F3 — Digital
Digital (94, 111, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3111, 19686, F3, 3, 17) (dual of [(19686, 3), 58947, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3111, 59058, F3, 17) (dual of [59058, 58947, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, 59059, F3, 17) (dual of [59059, 58948, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3111, 59059, F3, 17) (dual of [59059, 58948, 18]-code), using
- OOA 3-folding [i] based on linear OA(3111, 59058, F3, 17) (dual of [59058, 58947, 18]-code), using
(111−17, 111, 6840326)-Net in Base 3 — Upper bound on s
There is no (94, 111, 6840327)-net in base 3, because
- 1 times m-reduction [i] would yield (94, 110, 6840327)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 30432 537067 613089 891553 398838 555870 606067 516194 072433 > 3110 [i]