Best Known (213, 213+31, s)-Nets in Base 3
(213, 213+31, 35430)-Net over F3 — Constructive and digital
Digital (213, 244, 35430)-net over F3, using
- 32 times duplication [i] based on digital (211, 242, 35430)-net over F3, using
- net defined by OOA [i] based on linear OOA(3242, 35430, F3, 31, 31) (dual of [(35430, 31), 1098088, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3242, 531451, F3, 31) (dual of [531451, 531209, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 531454, F3, 31) (dual of [531454, 531212, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 531454, F3, 31) (dual of [531454, 531212, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3242, 531451, F3, 31) (dual of [531451, 531209, 32]-code), using
- net defined by OOA [i] based on linear OOA(3242, 35430, F3, 31, 31) (dual of [(35430, 31), 1098088, 32]-NRT-code), using
(213, 213+31, 132864)-Net over F3 — Digital
Digital (213, 244, 132864)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3244, 132864, F3, 4, 31) (dual of [(132864, 4), 531212, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3244, 531456, F3, 31) (dual of [531456, 531212, 32]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3242, 531454, F3, 31) (dual of [531454, 531212, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(30) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3242, 531454, F3, 31) (dual of [531454, 531212, 32]-code), using
- OOA 4-folding [i] based on linear OA(3244, 531456, F3, 31) (dual of [531456, 531212, 32]-code), using
(213, 213+31, large)-Net in Base 3 — Upper bound on s
There is no (213, 244, large)-net in base 3, because
- 29 times m-reduction [i] would yield (213, 215, large)-net in base 3, but