Best Known (83, 83+17, s)-Nets in Base 3
(83, 83+17, 2461)-Net over F3 — Constructive and digital
Digital (83, 100, 2461)-net over F3, using
- net defined by OOA [i] based on linear OOA(3100, 2461, F3, 17, 17) (dual of [(2461, 17), 41737, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3100, 19689, F3, 17) (dual of [19689, 19589, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3100, 19692, F3, 17) (dual of [19692, 19592, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3100, 19692, F3, 17) (dual of [19692, 19592, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3100, 19689, F3, 17) (dual of [19689, 19589, 18]-code), using
(83, 83+17, 6598)-Net over F3 — Digital
Digital (83, 100, 6598)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3100, 6598, F3, 2, 17) (dual of [(6598, 2), 13096, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3100, 9846, F3, 2, 17) (dual of [(9846, 2), 19592, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3100, 19692, F3, 17) (dual of [19692, 19592, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3100, 19692, F3, 17) (dual of [19692, 19592, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(3100, 9846, F3, 2, 17) (dual of [(9846, 2), 19592, 18]-NRT-code), using
(83, 83+17, 1510196)-Net in Base 3 — Upper bound on s
There is no (83, 100, 1510197)-net in base 3, because
- 1 times m-reduction [i] would yield (83, 99, 1510197)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 171793 023008 267050 681908 052782 106295 983159 418721 > 399 [i]