Best Known (83−9, 83, s)-Nets in Base 3
(83−9, 83, 398588)-Net over F3 — Constructive and digital
Digital (74, 83, 398588)-net over F3, using
- 32 times duplication [i] based on digital (72, 81, 398588)-net over F3, using
- net defined by OOA [i] based on linear OOA(381, 398588, F3, 9, 9) (dual of [(398588, 9), 3587211, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(381, 1594353, F3, 9) (dual of [1594353, 1594272, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(380, 1594352, F3, 9) (dual of [1594352, 1594272, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(379, 1594324, F3, 9) (dual of [1594324, 1594245, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(353, 1594324, F3, 7) (dual of [1594324, 1594271, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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 X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(380, 1594352, F3, 9) (dual of [1594352, 1594272, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(381, 1594353, F3, 9) (dual of [1594353, 1594272, 10]-code), using
- net defined by OOA [i] based on linear OOA(381, 398588, F3, 9, 9) (dual of [(398588, 9), 3587211, 10]-NRT-code), using
(83−9, 83, 797178)-Net over F3 — Digital
Digital (74, 83, 797178)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(383, 797178, F3, 2, 9) (dual of [(797178, 2), 1594273, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(383, 1594356, F3, 9) (dual of [1594356, 1594273, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(383, 1594357, F3, 9) (dual of [1594357, 1594274, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(380, 1594352, F3, 9) (dual of [1594352, 1594272, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(379, 1594324, F3, 9) (dual of [1594324, 1594245, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(353, 1594324, F3, 7) (dual of [1594324, 1594271, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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 X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(380, 1594354, F3, 7) (dual of [1594354, 1594274, 8]-code), using Gilbert–Varšamov bound and bm = 380 > Vbs−1(k−1) = 1 459994 126302 785938 500549 403273 431107 [i]
- linear OA(31, 3, F3, 1) (dual of [3, 2, 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 (see above)
- linear OA(380, 1594352, F3, 9) (dual of [1594352, 1594272, 10]-code), using
- construction X with Varšamov bound [i] based on
- discarding factors / shortening the dual code based on linear OA(383, 1594357, F3, 9) (dual of [1594357, 1594274, 10]-code), using
- OOA 2-folding [i] based on linear OA(383, 1594356, F3, 9) (dual of [1594356, 1594273, 10]-code), using
(83−9, 83, large)-Net in Base 3 — Upper bound on s
There is no (74, 83, large)-net in base 3, because
- 7 times m-reduction [i] would yield (74, 76, large)-net in base 3, but