Best Known (237−34, 237, s)-Nets in Base 4
(237−34, 237, 15423)-Net over F4 — Constructive and digital
Digital (203, 237, 15423)-net over F4, using
- net defined by OOA [i] based on linear OOA(4237, 15423, F4, 34, 34) (dual of [(15423, 34), 524145, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4237, 262191, F4, 34) (dual of [262191, 261954, 35]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4233, 262187, F4, 34) (dual of [262187, 261954, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4233, 262187, F4, 34) (dual of [262187, 261954, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4237, 262191, F4, 34) (dual of [262191, 261954, 35]-code), using
(237−34, 237, 131096)-Net over F4 — Digital
Digital (203, 237, 131096)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4237, 131096, F4, 2, 34) (dual of [(131096, 2), 261955, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4237, 262192, F4, 34) (dual of [262192, 261955, 35]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4233, 262187, F4, 34) (dual of [262187, 261954, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4233, 262188, F4, 30) (dual of [262188, 261955, 31]-code), using Gilbert–Varšamov bound and bm = 4233 > Vbs−1(k−1) = 106 916447 553568 232487 569708 261590 553420 588532 883435 996397 531101 270331 908787 734003 283649 197642 380681 850499 396318 906452 123122 897911 162837 331792 [i]
- linear OA(43, 4, F4, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,4) or 4-cap in PG(2,4)), using
- dual of repetition code with length 4 [i]
- Reed–Solomon code RS(1,4) [i]
- linear OA(4233, 262187, F4, 34) (dual of [262187, 261954, 35]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(4237, 262192, F4, 34) (dual of [262192, 261955, 35]-code), using
(237−34, 237, large)-Net in Base 4 — Upper bound on s
There is no (203, 237, large)-net in base 4, because
- 32 times m-reduction [i] would yield (203, 205, large)-net in base 4, but