Best Known (38, 38+11, s)-Nets in Base 7
(38, 38+11, 3364)-Net over F7 — Constructive and digital
Digital (38, 49, 3364)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 3364, F7, 11, 11) (dual of [(3364, 11), 36955, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(749, 16821, F7, 11) (dual of [16821, 16772, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 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
- discarding factors / shortening the dual code based on linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(749, 16821, F7, 11) (dual of [16821, 16772, 12]-code), using
(38, 38+11, 16825)-Net over F7 — Digital
Digital (38, 49, 16825)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 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
(38, 38+11, large)-Net in Base 7 — Upper bound on s
There is no (38, 49, large)-net in base 7, because
- 9 times m-reduction [i] would yield (38, 40, large)-net in base 7, but