Best Known (213−32, 213, s)-Nets in Base 3
(213−32, 213, 3691)-Net over F3 — Constructive and digital
Digital (181, 213, 3691)-net over F3, using
- 32 times duplication [i] based on digital (179, 211, 3691)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 3691, F3, 32, 32) (dual of [(3691, 32), 117901, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3211, 59056, F3, 32) (dual of [59056, 58845, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 59059, F3, 32) (dual of [59059, 58848, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 59059, F3, 32) (dual of [59059, 58848, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3211, 59056, F3, 32) (dual of [59056, 58845, 33]-code), using
- net defined by OOA [i] based on linear OOA(3211, 3691, F3, 32, 32) (dual of [(3691, 32), 117901, 33]-NRT-code), using
(213−32, 213, 19687)-Net over F3 — Digital
Digital (181, 213, 19687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3213, 19687, F3, 3, 32) (dual of [(19687, 3), 58848, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3213, 59061, F3, 32) (dual of [59061, 58848, 33]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3211, 59059, F3, 32) (dual of [59059, 58848, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(31) ⊂ Ce(30) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3211, 59059, F3, 32) (dual of [59059, 58848, 33]-code), using
- OOA 3-folding [i] based on linear OA(3213, 59061, F3, 32) (dual of [59061, 58848, 33]-code), using
(213−32, 213, 7641604)-Net in Base 3 — Upper bound on s
There is no (181, 213, 7641605)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 423475 013408 650960 950923 331216 934391 662023 302723 128938 219684 977474 440428 229021 153555 608110 992152 798657 > 3213 [i]