Best Known (173, 173+31, s)-Nets in Base 3
(173, 173+31, 3937)-Net over F3 — Constructive and digital
Digital (173, 204, 3937)-net over F3, using
- 32 times duplication [i] based on digital (171, 202, 3937)-net over F3, using
- net defined by OOA [i] based on linear OOA(3202, 3937, F3, 31, 31) (dual of [(3937, 31), 121845, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3202, 59056, F3, 31) (dual of [59056, 58854, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3202, 59056, F3, 31) (dual of [59056, 58854, 32]-code), using
- net defined by OOA [i] based on linear OOA(3202, 3937, F3, 31, 31) (dual of [(3937, 31), 121845, 32]-NRT-code), using
(173, 173+31, 19440)-Net over F3 — Digital
Digital (173, 204, 19440)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3204, 19440, F3, 3, 31) (dual of [(19440, 3), 58116, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3204, 19687, F3, 3, 31) (dual of [(19687, 3), 58857, 32]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3203, 19687, F3, 3, 31) (dual of [(19687, 3), 58858, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3203, 59061, F3, 31) (dual of [59061, 58858, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- OOA 3-folding [i] based on linear OA(3203, 59061, F3, 31) (dual of [59061, 58858, 32]-code), using
- 31 times duplication [i] based on linear OOA(3203, 19687, F3, 3, 31) (dual of [(19687, 3), 58858, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3204, 19687, F3, 3, 31) (dual of [(19687, 3), 58857, 32]-NRT-code), using
(173, 173+31, large)-Net in Base 3 — Upper bound on s
There is no (173, 204, large)-net in base 3, because
- 29 times m-reduction [i] would yield (173, 175, large)-net in base 3, but