Best Known (46, 46+7, s)-Nets in Base 4
(46, 46+7, 349532)-Net over F4 — Constructive and digital
Digital (46, 53, 349532)-net over F4, using
- 41 times duplication [i] based on digital (45, 52, 349532)-net over F4, using
- net defined by OOA [i] based on linear OOA(452, 349532, F4, 7, 7) (dual of [(349532, 7), 2446672, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(452, 1048597, F4, 7) (dual of [1048597, 1048545, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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 X applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(452, 1048597, F4, 7) (dual of [1048597, 1048545, 8]-code), using
- net defined by OOA [i] based on linear OOA(452, 349532, F4, 7, 7) (dual of [(349532, 7), 2446672, 8]-NRT-code), using
(46, 46+7, 1048600)-Net over F4 — Digital
Digital (46, 53, 1048600)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 1048600, F4, 7) (dual of [1048600, 1048547, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(452, 1048598, F4, 7) (dual of [1048598, 1048546, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 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(6) ⊂ Ce(4) [i] based on
- linear OA(452, 1048599, F4, 6) (dual of [1048599, 1048547, 7]-code), using Gilbert–Varšamov bound and bm = 452 > Vbs−1(k−1) = 2 567241 362716 714849 103439 157771 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(452, 1048598, F4, 7) (dual of [1048598, 1048546, 8]-code), using
- construction X with Varšamov bound [i] based on
(46, 46+7, large)-Net in Base 4 — Upper bound on s
There is no (46, 53, large)-net in base 4, because
- 5 times m-reduction [i] would yield (46, 48, large)-net in base 4, but