Best Known (232−29, 232, s)-Nets in Base 3
(232−29, 232, 37961)-Net over F3 — Constructive and digital
Digital (203, 232, 37961)-net over F3, using
- 31 times duplication [i] based on digital (202, 231, 37961)-net over F3, using
- net defined by OOA [i] based on linear OOA(3231, 37961, F3, 29, 29) (dual of [(37961, 29), 1100638, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3231, 531455, F3, 29) (dual of [531455, 531224, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3229, 531453, F3, 29) (dual of [531453, 531224, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3229, 531453, F3, 29) (dual of [531453, 531224, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3231, 531455, F3, 29) (dual of [531455, 531224, 30]-code), using
- net defined by OOA [i] based on linear OOA(3231, 37961, F3, 29, 29) (dual of [(37961, 29), 1100638, 30]-NRT-code), using
(232−29, 232, 132864)-Net over F3 — Digital
Digital (203, 232, 132864)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3232, 132864, F3, 4, 29) (dual of [(132864, 4), 531224, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3232, 531456, F3, 29) (dual of [531456, 531224, 30]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3229, 531453, F3, 29) (dual of [531453, 531224, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3229, 531453, F3, 29) (dual of [531453, 531224, 30]-code), using
- OOA 4-folding [i] based on linear OA(3232, 531456, F3, 29) (dual of [531456, 531224, 30]-code), using
(232−29, 232, large)-Net in Base 3 — Upper bound on s
There is no (203, 232, large)-net in base 3, because
- 27 times m-reduction [i] would yield (203, 205, large)-net in base 3, but