Best Known (52, 52+11, s)-Nets in Base 7
(52, 52+11, 47060)-Net over F7 — Constructive and digital
Digital (52, 63, 47060)-net over F7, using
- 71 times duplication [i] based on digital (51, 62, 47060)-net over F7, using
- net defined by OOA [i] based on linear OOA(762, 47060, F7, 11, 11) (dual of [(47060, 11), 517598, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(762, 235301, F7, 11) (dual of [235301, 235239, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 235304, F7, 11) (dual of [235304, 235242, 12]-code), using
- trace code [i] based on linear OA(4931, 117652, F49, 11) (dual of [117652, 117621, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(4931, 117649, F49, 11) (dual of [117649, 117618, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(4931, 117652, F49, 11) (dual of [117652, 117621, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 235304, F7, 11) (dual of [235304, 235242, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(762, 235301, F7, 11) (dual of [235301, 235239, 12]-code), using
- net defined by OOA [i] based on linear OOA(762, 47060, F7, 11, 11) (dual of [(47060, 11), 517598, 12]-NRT-code), using
(52, 52+11, 235306)-Net over F7 — Digital
Digital (52, 63, 235306)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(763, 235306, F7, 11) (dual of [235306, 235243, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(762, 235304, F7, 11) (dual of [235304, 235242, 12]-code), using
- trace code [i] based on linear OA(4931, 117652, F49, 11) (dual of [117652, 117621, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(4931, 117649, F49, 11) (dual of [117649, 117618, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(4931, 117652, F49, 11) (dual of [117652, 117621, 12]-code), using
- linear OA(762, 235305, F7, 10) (dual of [235305, 235243, 11]-code), using Gilbert–Varšamov bound and bm = 762 > Vbs−1(k−1) = 61 404236 191426 512334 124678 402318 203522 746040 359361 [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(762, 235304, F7, 11) (dual of [235304, 235242, 12]-code), using
- construction X with Varšamov bound [i] based on
(52, 52+11, large)-Net in Base 7 — Upper bound on s
There is no (52, 63, large)-net in base 7, because
- 9 times m-reduction [i] would yield (52, 54, large)-net in base 7, but