Best Known (110−14, 110, s)-Nets in Base 3
(110−14, 110, 75922)-Net over F3 — Constructive and digital
Digital (96, 110, 75922)-net over F3, using
- net defined by OOA [i] based on linear OOA(3110, 75922, F3, 14, 14) (dual of [(75922, 14), 1062798, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3110, 531454, F3, 14) (dual of [531454, 531344, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3110, 531455, F3, 14) (dual of [531455, 531345, 15]-code), using
- construction X4 applied to C([0,13]) ⊂ C([1,12]) [i] based on
- linear OA(3109, 531440, F3, 14) (dual of [531440, 531331, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(396, 531440, F3, 12) (dual of [531440, 531344, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 X4 applied to C([0,13]) ⊂ C([1,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3110, 531455, F3, 14) (dual of [531455, 531345, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3110, 531454, F3, 14) (dual of [531454, 531344, 15]-code), using
(110−14, 110, 177151)-Net over F3 — Digital
Digital (96, 110, 177151)-net over F3, using
- 31 times duplication [i] based on digital (95, 109, 177151)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3109, 177151, F3, 3, 14) (dual of [(177151, 3), 531344, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3109, 531453, F3, 14) (dual of [531453, 531344, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(3109, 531441, F3, 14) (dual of [531441, 531332, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 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(13) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(3109, 531453, F3, 14) (dual of [531453, 531344, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3109, 177151, F3, 3, 14) (dual of [(177151, 3), 531344, 15]-NRT-code), using
(110−14, 110, large)-Net in Base 3 — Upper bound on s
There is no (96, 110, large)-net in base 3, because
- 12 times m-reduction [i] would yield (96, 98, large)-net in base 3, but