Best Known (228−32, 228, s)-Nets in Base 4
(228−32, 228, 16387)-Net over F4 — Constructive and digital
Digital (196, 228, 16387)-net over F4, using
- 1 times m-reduction [i] based on digital (196, 229, 16387)-net over F4, using
- net defined by OOA [i] based on linear OOA(4229, 16387, F4, 33, 33) (dual of [(16387, 33), 540542, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4229, 262193, F4, 33) (dual of [262193, 261964, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4227, 262191, F4, 33) (dual of [262191, 261964, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4181, 262145, F4, 27) (dual of [262145, 261964, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(410, 46, F4, 5) (dual of [46, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4227, 262191, F4, 33) (dual of [262191, 261964, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4229, 262193, F4, 33) (dual of [262193, 261964, 34]-code), using
- net defined by OOA [i] based on linear OOA(4229, 16387, F4, 33, 33) (dual of [(16387, 33), 540542, 34]-NRT-code), using
(228−32, 228, 144272)-Net over F4 — Digital
Digital (196, 228, 144272)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4228, 144272, F4, 32) (dual of [144272, 144044, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4228, 262200, F4, 32) (dual of [262200, 261972, 33]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4227, 262199, F4, 32) (dual of [262199, 261972, 33]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4227, 262199, F4, 32) (dual of [262199, 261972, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4228, 262200, F4, 32) (dual of [262200, 261972, 33]-code), using
(228−32, 228, large)-Net in Base 4 — Upper bound on s
There is no (196, 228, large)-net in base 4, because
- 30 times m-reduction [i] would yield (196, 198, large)-net in base 4, but