Best Known (113−16, 113, s)-Nets in Base 3
(113−16, 113, 22145)-Net over F3 — Constructive and digital
Digital (97, 113, 22145)-net over F3, using
- net defined by OOA [i] based on linear OOA(3113, 22145, F3, 16, 16) (dual of [(22145, 16), 354207, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3113, 177160, F3, 16) (dual of [177160, 177047, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3112, 177159, F3, 16) (dual of [177159, 177047, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(3112, 177159, F3, 16) (dual of [177159, 177047, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3113, 177160, F3, 16) (dual of [177160, 177047, 17]-code), using
(113−16, 113, 59053)-Net over F3 — Digital
Digital (97, 113, 59053)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3113, 59053, F3, 3, 16) (dual of [(59053, 3), 177046, 17]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3112, 59053, F3, 3, 16) (dual of [(59053, 3), 177047, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3112, 177159, F3, 16) (dual of [177159, 177047, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- OOA 3-folding [i] based on linear OA(3112, 177159, F3, 16) (dual of [177159, 177047, 17]-code), using
- 31 times duplication [i] based on linear OOA(3112, 59053, F3, 3, 16) (dual of [(59053, 3), 177047, 17]-NRT-code), using
(113−16, 113, large)-Net in Base 3 — Upper bound on s
There is no (97, 113, large)-net in base 3, because
- 14 times m-reduction [i] would yield (97, 99, large)-net in base 3, but