Best Known (39−6, 39, s)-Nets in Base 4
(39−6, 39, 87385)-Net over F4 — Constructive and digital
Digital (33, 39, 87385)-net over F4, using
- 41 times duplication [i] based on digital (32, 38, 87385)-net over F4, using
- net defined by OOA [i] based on linear OOA(438, 87385, F4, 6, 6) (dual of [(87385, 6), 524272, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(438, 262155, F4, 6) (dual of [262155, 262117, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(410, 11, F4, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
- dual of repetition code with length 11 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(438, 262155, F4, 6) (dual of [262155, 262117, 7]-code), using
- net defined by OOA [i] based on linear OOA(438, 87385, F4, 6, 6) (dual of [(87385, 6), 524272, 7]-NRT-code), using
(39−6, 39, 262157)-Net over F4 — Digital
Digital (33, 39, 262157)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(439, 262157, F4, 6) (dual of [262157, 262118, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(437, 262153, F4, 6) (dual of [262153, 262116, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(437, 262155, F4, 5) (dual of [262155, 262118, 6]-code), using Gilbert–Varšamov bound and bm = 437 > Vbs−1(k−1) = 15940 135205 329475 686606 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s (see above)
- linear OA(437, 262153, F4, 6) (dual of [262153, 262116, 7]-code), using
- construction X with Varšamov bound [i] based on
(39−6, 39, large)-Net in Base 4 — Upper bound on s
There is no (33, 39, large)-net in base 4, because
- 4 times m-reduction [i] would yield (33, 35, large)-net in base 4, but