Best Known (212, 212+30, s)-Nets in Base 3
(212, 212+30, 35431)-Net over F3 — Constructive and digital
Digital (212, 242, 35431)-net over F3, using
- net defined by OOA [i] based on linear OOA(3242, 35431, F3, 30, 30) (dual of [(35431, 30), 1062688, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(3242, 531465, F3, 30) (dual of [531465, 531223, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 531466, F3, 30) (dual of [531466, 531224, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31, 25, F3, 1) (dual of [25, 24, 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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 531466, F3, 30) (dual of [531466, 531224, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(3242, 531465, F3, 30) (dual of [531465, 531223, 31]-code), using
(212, 212+30, 132866)-Net over F3 — Digital
Digital (212, 242, 132866)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3242, 132866, F3, 4, 30) (dual of [(132866, 4), 531222, 31]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3242, 531464, F3, 30) (dual of [531464, 531222, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 531466, F3, 30) (dual of [531466, 531224, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31, 25, F3, 1) (dual of [25, 24, 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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 531466, F3, 30) (dual of [531466, 531224, 31]-code), using
- OOA 4-folding [i] based on linear OA(3242, 531464, F3, 30) (dual of [531464, 531222, 31]-code), using
(212, 212+30, large)-Net in Base 3 — Upper bound on s
There is no (212, 242, large)-net in base 3, because
- 28 times m-reduction [i] would yield (212, 214, large)-net in base 3, but