Best Known (53, 53+13, s)-Nets in Base 3
(53, 53+13, 1094)-Net over F3 — Constructive and digital
Digital (53, 66, 1094)-net over F3, using
- net defined by OOA [i] based on linear OOA(366, 1094, F3, 13, 13) (dual of [(1094, 13), 14156, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(366, 6565, F3, 13) (dual of [6565, 6499, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(366, 6570, F3, 13) (dual of [6570, 6504, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(366, 6570, F3, 13) (dual of [6570, 6504, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(366, 6565, F3, 13) (dual of [6565, 6499, 14]-code), using
(53, 53+13, 2553)-Net over F3 — Digital
Digital (53, 66, 2553)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(366, 2553, F3, 2, 13) (dual of [(2553, 2), 5040, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(366, 3285, F3, 2, 13) (dual of [(3285, 2), 6504, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(366, 6570, F3, 13) (dual of [6570, 6504, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 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(12) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(366, 6570, F3, 13) (dual of [6570, 6504, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(366, 3285, F3, 2, 13) (dual of [(3285, 2), 6504, 14]-NRT-code), using
(53, 53+13, 220797)-Net in Base 3 — Upper bound on s
There is no (53, 66, 220798)-net in base 3, because
- 1 times m-reduction [i] would yield (53, 65, 220798)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10 301090 343411 098081 272144 490277 > 365 [i]