Best Known (91−17, 91, s)-Nets in Base 3
(91−17, 91, 821)-Net over F3 — Constructive and digital
Digital (74, 91, 821)-net over F3, using
- 32 times duplication [i] based on digital (72, 89, 821)-net over F3, using
- net defined by OOA [i] based on linear OOA(389, 821, F3, 17, 17) (dual of [(821, 17), 13868, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(389, 6569, F3, 17) (dual of [6569, 6480, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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(16) ⊂ Ce(15) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(389, 6569, F3, 17) (dual of [6569, 6480, 18]-code), using
- net defined by OOA [i] based on linear OOA(389, 821, F3, 17, 17) (dual of [(821, 17), 13868, 18]-NRT-code), using
(91−17, 91, 3250)-Net over F3 — Digital
Digital (74, 91, 3250)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(391, 3250, F3, 2, 17) (dual of [(3250, 2), 6409, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(391, 3286, F3, 2, 17) (dual of [(3286, 2), 6481, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(391, 6572, F3, 17) (dual of [6572, 6481, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(391, 6572, F3, 17) (dual of [6572, 6481, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(391, 3286, F3, 2, 17) (dual of [(3286, 2), 6481, 18]-NRT-code), using
(91−17, 91, 438800)-Net in Base 3 — Upper bound on s
There is no (74, 91, 438801)-net in base 3, because
- 1 times m-reduction [i] would yield (74, 90, 438801)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8 728099 605653 485413 684475 855559 953390 916001 > 390 [i]