Best Known (79, 79+19, s)-Nets in Base 3
(79, 79+19, 729)-Net over F3 — Constructive and digital
Digital (79, 98, 729)-net over F3, using
- 31 times duplication [i] based on digital (78, 97, 729)-net over F3, using
- net defined by OOA [i] based on linear OOA(397, 729, F3, 19, 19) (dual of [(729, 19), 13754, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- net defined by OOA [i] based on linear OOA(397, 729, F3, 19, 19) (dual of [(729, 19), 13754, 20]-NRT-code), using
(79, 79+19, 2464)-Net over F3 — Digital
Digital (79, 98, 2464)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 2464, F3, 2, 19) (dual of [(2464, 2), 4830, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(398, 3285, F3, 2, 19) (dual of [(3285, 2), 6472, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(398, 6570, F3, 19) (dual of [6570, 6472, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(398, 6570, F3, 19) (dual of [6570, 6472, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(398, 3285, F3, 2, 19) (dual of [(3285, 2), 6472, 20]-NRT-code), using
(79, 79+19, 287750)-Net in Base 3 — Upper bound on s
There is no (79, 98, 287751)-net in base 3, because
- 1 times m-reduction [i] would yield (79, 97, 287751)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 19088 594769 468584 800768 914669 272430 013895 030783 > 397 [i]