Best Known (73−11, 73, s)-Nets in Base 4
(73−11, 73, 52430)-Net over F4 — Constructive and digital
Digital (62, 73, 52430)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 52430, F4, 11, 11) (dual of [(52430, 11), 576657, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(473, 262151, F4, 11) (dual of [262151, 262078, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 262153, F4, 11) (dual of [262153, 262080, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(473, 262144, F4, 11) (dual of [262144, 262071, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 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
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(473, 262153, F4, 11) (dual of [262153, 262080, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(473, 262151, F4, 11) (dual of [262151, 262078, 12]-code), using
(73−11, 73, 131076)-Net over F4 — Digital
Digital (62, 73, 131076)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(473, 131076, F4, 2, 11) (dual of [(131076, 2), 262079, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(473, 262152, F4, 11) (dual of [262152, 262079, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 262153, F4, 11) (dual of [262153, 262080, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(473, 262144, F4, 11) (dual of [262144, 262071, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 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
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(473, 262153, F4, 11) (dual of [262153, 262080, 12]-code), using
- OOA 2-folding [i] based on linear OA(473, 262152, F4, 11) (dual of [262152, 262079, 12]-code), using
(73−11, 73, large)-Net in Base 4 — Upper bound on s
There is no (62, 73, large)-net in base 4, because
- 9 times m-reduction [i] would yield (62, 64, large)-net in base 4, but