Best Known (35, 44, s)-Nets in Base 4
(35, 44, 4097)-Net over F4 — Constructive and digital
Digital (35, 44, 4097)-net over F4, using
- net defined by OOA [i] based on linear OOA(444, 4097, F4, 9, 9) (dual of [(4097, 9), 36829, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(444, 16389, F4, 9) (dual of [16389, 16345, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(444, 16392, F4, 9) (dual of [16392, 16348, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 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(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(444, 16392, F4, 9) (dual of [16392, 16348, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(444, 16389, F4, 9) (dual of [16389, 16345, 10]-code), using
(35, 44, 8196)-Net over F4 — Digital
Digital (35, 44, 8196)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(444, 8196, F4, 2, 9) (dual of [(8196, 2), 16348, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(444, 16392, F4, 9) (dual of [16392, 16348, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 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(8) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(444, 16392, F4, 9) (dual of [16392, 16348, 10]-code), using
(35, 44, 2188144)-Net in Base 4 — Upper bound on s
There is no (35, 44, 2188145)-net in base 4, because
- 1 times m-reduction [i] would yield (35, 43, 2188145)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 77 371393 206774 103757 185156 > 443 [i]