MinT - Best Known (131−10, 131, s)-Nets in Base 5

Best Known (131−10, 131, s)-Nets in Base 5

(131−10, 131, 4332032)-Net over F₅ — Constructive and digital

Digital (121, 131, 4332032)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (34, 39, 976592)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5³⁹, 976592, F₅, 6, 5) (dual of [(976592, 6), 5859513, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(5³⁹, 976593, F₅, 2, 5) (dual of [(976593, 2), 1953147, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(5³, 31, F₅, 2, 2) (dual of [(31, 2), 59, 3]-NRT-code), using
        appending kth column [i] based on linear OA(5³, 31, F₅, 2) (dual of [31, 28, 3]-code), using
        Hamming code H(3,5) [i]
        linear OOA(5³⁶, 976562, F₅, 2, 5) (dual of [(976562, 2), 1953088, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(5³⁶, 1953124, F₅, 5) (dual of [1953124, 1953088, 6]-code), using
        1 times truncation [i] based on linear OA(5³⁷, 1953125, F₅, 6) (dual of [1953125, 1953088, 7]-code), using
        an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1953124Â = 5⁹âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
2. digital (82, 92, 3355440)-net over F₅, using
  - trace code for nets [i] based on digital (36, 46, 1677720)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁴⁶, 1677720, F₂₅, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(25⁴⁶, 8388600, F₂₅, 10) (dual of [8388600, 8388554, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴⁶, large, F₂₅, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(131−10, 131, large)-Net over F₅ — Digital

Digital (121, 131, large)-net over F₅, using

6 times m-reduction [i] based on digital (121, 137, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³⁷, large, F₅, 16) (dual of [large, large−137, 17]-code), using
  - 16 times code embedding in larger space [i] based on linear OA(5¹²¹, large, F₅, 16) (dual of [large, large−121, 17]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(131−10, 131, large)-Net in Base 5 — Upper bound on s

There is no (121, 131, large)-net in base 5, because

8 times m-reduction [i] would yield (121, 123, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (131−10, 131, s)-Nets in Base 5

(131−10, 131, 4332032)-Net over F5 — Constructive and digital

(131−10, 131, large)-Net over F5 — Digital

(131−10, 131, large)-Net in Base 5 — Upper bound on s

(131−10, 131, 4332032)-Net over F₅ — Constructive and digital

(131−10, 131, large)-Net over F₅ — Digital