Best Known (36, 36+9, s)-Nets in Base 4
(36, 36+9, 4098)-Net over F4 — Constructive and digital
Digital (36, 45, 4098)-net over F4, using
- net defined by OOA [i] based on linear OOA(445, 4098, F4, 9, 9) (dual of [(4098, 9), 36837, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(445, 16393, F4, 9) (dual of [16393, 16348, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(445, 16394, F4, 9) (dual of [16394, 16349, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [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(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(445, 16394, F4, 9) (dual of [16394, 16349, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(445, 16393, F4, 9) (dual of [16393, 16348, 10]-code), using
(36, 36+9, 8197)-Net over F4 — Digital
Digital (36, 45, 8197)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(445, 8197, F4, 2, 9) (dual of [(8197, 2), 16349, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(445, 16394, F4, 9) (dual of [16394, 16349, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [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(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(445, 16394, F4, 9) (dual of [16394, 16349, 10]-code), using
(36, 36+9, 3094504)-Net in Base 4 — Upper bound on s
There is no (36, 45, 3094505)-net in base 4, because
- 1 times m-reduction [i] would yield (36, 44, 3094505)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 309 485369 187126 540011 486446 > 444 [i]