Best Known (48, 48+11, s)-Nets in Base 7
(48, 48+11, 23534)-Net over F7 — Constructive and digital
Digital (48, 59, 23534)-net over F7, using
- net defined by OOA [i] based on linear OOA(759, 23534, F7, 11, 11) (dual of [(23534, 11), 258815, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(759, 117671, F7, 11) (dual of [117671, 117612, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(758, 117670, F7, 11) (dual of [117670, 117612, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 21, F7, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(758, 117670, F7, 11) (dual of [117670, 117612, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(759, 117671, F7, 11) (dual of [117671, 117612, 12]-code), using
(48, 48+11, 117672)-Net over F7 — Digital
Digital (48, 59, 117672)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(759, 117672, F7, 11) (dual of [117672, 117613, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(758, 117670, F7, 11) (dual of [117670, 117612, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 21, F7, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(758, 117671, F7, 10) (dual of [117671, 117613, 11]-code), using Gilbert–Varšamov bound and bm = 758 > Vbs−1(k−1) = 120077 784637 037613 092390 119919 115879 013522 782865 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(758, 117670, F7, 11) (dual of [117670, 117612, 12]-code), using
- construction X with Varšamov bound [i] based on
(48, 48+11, large)-Net in Base 7 — Upper bound on s
There is no (48, 59, large)-net in base 7, because
- 9 times m-reduction [i] would yield (48, 50, large)-net in base 7, but