Best Known (113, 113+15, s)-Nets in Base 3
(113, 113+15, 75926)-Net over F3 — Constructive and digital
Digital (113, 128, 75926)-net over F3, using
- 31 times duplication [i] based on digital (112, 127, 75926)-net over F3, using
- net defined by OOA [i] based on linear OOA(3127, 75926, F3, 15, 15) (dual of [(75926, 15), 1138763, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3127, 531483, F3, 15) (dual of [531483, 531356, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(3127, 531483, F3, 15) (dual of [531483, 531356, 16]-code), using
- net defined by OOA [i] based on linear OOA(3127, 75926, F3, 15, 15) (dual of [(75926, 15), 1138763, 16]-NRT-code), using
(113, 113+15, 265742)-Net over F3 — Digital
Digital (113, 128, 265742)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3128, 265742, F3, 2, 15) (dual of [(265742, 2), 531356, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3128, 531484, F3, 15) (dual of [531484, 531356, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3127, 531483, F3, 15) (dual of [531483, 531356, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3127, 531483, F3, 15) (dual of [531483, 531356, 16]-code), using
- OOA 2-folding [i] based on linear OA(3128, 531484, F3, 15) (dual of [531484, 531356, 16]-code), using
(113, 113+15, large)-Net in Base 3 — Upper bound on s
There is no (113, 128, large)-net in base 3, because
- 13 times m-reduction [i] would yield (113, 115, large)-net in base 3, but