Best Known (72, 72+10, s)-Nets in Base 3
(72, 72+10, 318868)-Net over F3 — Constructive and digital
Digital (72, 82, 318868)-net over F3, using
- net defined by OOA [i] based on linear OOA(382, 318868, F3, 10, 10) (dual of [(318868, 10), 3188598, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(382, 1594340, F3, 10) (dual of [1594340, 1594258, 11]-code), using
- 2 times code embedding in larger space [i] based on linear OA(380, 1594338, F3, 10) (dual of [1594338, 1594258, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 Ce(9) ⊂ Ce(7) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(380, 1594338, F3, 10) (dual of [1594338, 1594258, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(382, 1594340, F3, 10) (dual of [1594340, 1594258, 11]-code), using
(72, 72+10, 531446)-Net over F3 — Digital
Digital (72, 82, 531446)-net over F3, using
- 32 times duplication [i] based on digital (70, 80, 531446)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(380, 531446, F3, 3, 10) (dual of [(531446, 3), 1594258, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(380, 1594338, F3, 10) (dual of [1594338, 1594258, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 Ce(9) ⊂ Ce(7) [i] based on
- OOA 3-folding [i] based on linear OA(380, 1594338, F3, 10) (dual of [1594338, 1594258, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(380, 531446, F3, 3, 10) (dual of [(531446, 3), 1594258, 11]-NRT-code), using
(72, 72+10, large)-Net in Base 3 — Upper bound on s
There is no (72, 82, large)-net in base 3, because
- 8 times m-reduction [i] would yield (72, 74, large)-net in base 3, but