Best Known (54, 54+11, s)-Nets in Base 3
(54, 54+11, 3938)-Net over F3 — Constructive and digital
Digital (54, 65, 3938)-net over F3, using
- 31 times duplication [i] based on digital (53, 64, 3938)-net over F3, using
- net defined by OOA [i] based on linear OOA(364, 3938, F3, 11, 11) (dual of [(3938, 11), 43254, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(364, 19691, F3, 11) (dual of [19691, 19627, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(364, 19692, F3, 11) (dual of [19692, 19628, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(364, 19683, F3, 11) (dual of [19683, 19619, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(364, 19692, F3, 11) (dual of [19692, 19628, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(364, 19691, F3, 11) (dual of [19691, 19627, 12]-code), using
- net defined by OOA [i] based on linear OOA(364, 3938, F3, 11, 11) (dual of [(3938, 11), 43254, 12]-NRT-code), using
(54, 54+11, 9847)-Net over F3 — Digital
Digital (54, 65, 9847)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(365, 9847, F3, 2, 11) (dual of [(9847, 2), 19629, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(365, 19694, F3, 11) (dual of [19694, 19629, 12]-code), using
- construction X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- linear OA(364, 19682, F3, 11) (dual of [19682, 19618, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(354, 19682, F3, 9) (dual of [19682, 19628, 10]-code), using the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [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 X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- OOA 2-folding [i] based on linear OA(365, 19694, F3, 11) (dual of [19694, 19629, 12]-code), using
(54, 54+11, 1667082)-Net in Base 3 — Upper bound on s
There is no (54, 65, 1667083)-net in base 3, because
- 1 times m-reduction [i] would yield (54, 64, 1667083)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 433687 029177 617558 659097 916495 > 364 [i]