Best Known (59−11, 59, s)-Nets in Base 3
(59−11, 59, 1314)-Net over F3 — Constructive and digital
Digital (48, 59, 1314)-net over F3, using
- net defined by OOA [i] based on linear OOA(359, 1314, F3, 11, 11) (dual of [(1314, 11), 14395, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(359, 6571, F3, 11) (dual of [6571, 6512, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(359, 6572, F3, 11) (dual of [6572, 6513, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(10) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(359, 6572, F3, 11) (dual of [6572, 6513, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(359, 6571, F3, 11) (dual of [6571, 6512, 12]-code), using
(59−11, 59, 3286)-Net over F3 — Digital
Digital (48, 59, 3286)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(359, 3286, F3, 2, 11) (dual of [(3286, 2), 6513, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(359, 6572, F3, 11) (dual of [6572, 6513, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(10) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(359, 6572, F3, 11) (dual of [6572, 6513, 12]-code), using
(59−11, 59, 446075)-Net in Base 3 — Upper bound on s
There is no (48, 59, 446076)-net in base 3, because
- 1 times m-reduction [i] would yield (48, 58, 446076)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4710 152045 401238 238663 238873 > 358 [i]