Best Known (72−11, 72, s)-Nets in Base 3
(72−11, 72, 11812)-Net over F3 — Constructive and digital
Digital (61, 72, 11812)-net over F3, using
- net defined by OOA [i] based on linear OOA(372, 11812, F3, 11, 11) (dual of [(11812, 11), 129860, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(372, 59061, F3, 11) (dual of [59061, 58989, 12]-code), using
- construction X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- linear OA(371, 59048, F3, 11) (dual of [59048, 58977, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(360, 59048, F3, 9) (dual of [59048, 58988, 10]-code), using 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(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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 X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(372, 59061, F3, 11) (dual of [59061, 58989, 12]-code), using
(72−11, 72, 28143)-Net over F3 — Digital
Digital (61, 72, 28143)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(372, 28143, F3, 2, 11) (dual of [(28143, 2), 56214, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(372, 29530, F3, 2, 11) (dual of [(29530, 2), 58988, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(372, 59060, F3, 11) (dual of [59060, 58988, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 59061, F3, 11) (dual of [59061, 58989, 12]-code), using
- construction X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- linear OA(371, 59048, F3, 11) (dual of [59048, 58977, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(360, 59048, F3, 9) (dual of [59048, 58988, 10]-code), using 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(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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 X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(372, 59061, F3, 11) (dual of [59061, 58989, 12]-code), using
- OOA 2-folding [i] based on linear OA(372, 59060, F3, 11) (dual of [59060, 58988, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(372, 29530, F3, 2, 11) (dual of [(29530, 2), 58988, 12]-NRT-code), using
(72−11, 72, 7761181)-Net in Base 3 — Upper bound on s
There is no (61, 72, 7761182)-net in base 3, because
- 1 times m-reduction [i] would yield (61, 71, 7761182)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7509 470753 032459 536720 348347 329365 > 371 [i]