Best Known (247−34, 247, s)-Nets in Base 3
(247−34, 247, 10421)-Net over F3 — Constructive and digital
Digital (213, 247, 10421)-net over F3, using
- 33 times duplication [i] based on digital (210, 244, 10421)-net over F3, using
- net defined by OOA [i] based on linear OOA(3244, 10421, F3, 34, 34) (dual of [(10421, 34), 354070, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(3244, 177157, F3, 34) (dual of [177157, 176913, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 177159, F3, 34) (dual of [177159, 176915, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3243, 177147, F3, 34) (dual of [177147, 176904, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 177159, F3, 34) (dual of [177159, 176915, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(3244, 177157, F3, 34) (dual of [177157, 176913, 35]-code), using
- net defined by OOA [i] based on linear OOA(3244, 10421, F3, 34, 34) (dual of [(10421, 34), 354070, 35]-NRT-code), using
(247−34, 247, 45716)-Net over F3 — Digital
Digital (213, 247, 45716)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3247, 45716, F3, 3, 34) (dual of [(45716, 3), 136901, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3247, 59057, F3, 3, 34) (dual of [(59057, 3), 176924, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3247, 177171, F3, 34) (dual of [177171, 176924, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3247, 177173, F3, 34) (dual of [177173, 176926, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(3243, 177147, F3, 34) (dual of [177147, 176904, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3247, 177173, F3, 34) (dual of [177173, 176926, 35]-code), using
- OOA 3-folding [i] based on linear OA(3247, 177171, F3, 34) (dual of [177171, 176924, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3247, 59057, F3, 3, 34) (dual of [(59057, 3), 176924, 35]-NRT-code), using
(247−34, 247, large)-Net in Base 3 — Upper bound on s
There is no (213, 247, large)-net in base 3, because
- 32 times m-reduction [i] would yield (213, 215, large)-net in base 3, but