MinT - Best Known (142−23, 142, s)-Nets in Base 5

Best Known (142−23, 142, s)-Nets in Base 5

(142−23, 142, 7120)-Net over F₅ — Constructive and digital

Digital (119, 142, 7120)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (104, 127, 7102)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹²⁷, 7102, F₅, 23, 23) (dual of [(7102, 23), 163219, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(5¹²⁷, 78123, F₅, 23) (dual of [78123, 77996, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹²⁷, 78125, F₅, 23) (dual of [78125, 77998, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(142−23, 142, 78185)-Net over F₅ — Digital

Digital (119, 142, 78185)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴², 78185, F₅, 23) (dual of [78185, 78043, 24]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(5¹³⁹, 78179, F₅, 23) (dual of [78179, 78040, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(15) [i] based on
      1. linear OA(5¹²⁷, 78125, F₅, 23) (dual of [78125, 77998, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(5⁸⁵, 78125, F₅, 16) (dual of [78125, 78040, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(5¹², 54, F₅, 6) (dual of [54, 42, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(5¹², 62, F₅, 6) (dual of [62, 50, 7]-code), using
        the cyclic code C(A) with length 62Â | 5³âˆ’1, defining set AÂ =Â {4,8,11,17}, and minimum distance dÂ â‰¥ |{8,11,14,…,23}|+1Â =Â 7 (BCH-bound) [i]
  2. linear OA(5¹³⁹, 78182, F₅, 22) (dual of [78182, 78043, 23]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 5¹³⁹Â > V_b^sâˆ’1(kâˆ’1)Â = 488544 437075 719871 198631 007757 800155 490924 422537 884517 481217 081023 098643 576526 826556 629305 555125 [i]
  3. linear OA(5⁰, 3, F₅, 0) (dual of [3, 3, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(142−23, 142, large)-Net in Base 5 — Upper bound on s

There is no (119, 142, large)-net in base 5, because

21 times m-reduction [i] would yield (119, 121, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (142−23, 142, s)-Nets in Base 5

(142−23, 142, 7120)-Net over F5 — Constructive and digital

(142−23, 142, 78185)-Net over F5 — Digital

(142−23, 142, large)-Net in Base 5 — Upper bound on s

(142−23, 142, 7120)-Net over F₅ — Constructive and digital

(142−23, 142, 78185)-Net over F₅ — Digital