Best Known (52, 62, s)-Nets in Base 3
(52, 62, 11812)-Net over F3 — Constructive and digital
Digital (52, 62, 11812)-net over F3, using
- net defined by OOA [i] based on linear OOA(362, 11812, F3, 10, 10) (dual of [(11812, 10), 118058, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(362, 59060, F3, 10) (dual of [59060, 58998, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(362, 59060, F3, 10) (dual of [59060, 58998, 11]-code), using
(52, 62, 20767)-Net over F3 — Digital
Digital (52, 62, 20767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(362, 20767, F3, 2, 10) (dual of [(20767, 2), 41472, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(362, 29530, F3, 2, 10) (dual of [(29530, 2), 58998, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(362, 59060, F3, 10) (dual of [59060, 58998, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(9) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(362, 59060, F3, 10) (dual of [59060, 58998, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(362, 29530, F3, 2, 10) (dual of [(29530, 2), 58998, 11]-NRT-code), using
(52, 62, 1074256)-Net in Base 3 — Upper bound on s
There is no (52, 62, 1074257)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 381521 300514 993294 883873 698195 > 362 [i]