@article{Xu2025, 
author = {Xiaofan Xu and Zongbing Lin},
title = {On the deep holes of a class of Cauchy codes},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {10},
pages = {23534-23546},
keywords = {Cauchy code, deep hole, Lagrange interpolation polynomial},
url = {https://www.sciopen.com/article/10.3934/math.20251045},
doi = {10.3934/math.20251045},
abstract = {Suppose              F        q   is a finite field having an odd characteristic. Let    D  =  {      x    1    ,  ⋯  ,      x          n      −      1        ,  ∞  }, where    {      x    1    ,  ⋯  ,      x          n      −      1        }  ⊊            F        q  . Assume that    k is an integer with    2  ≤  k  <  n. We show that    u  (  D  ) is a deep hole of Cauchy code    C  (  D  ,  k  ) if    u  (  x  )  =  λ  (  x  −  δ      )          q      −      2        +  ν      x          k      −      1        +      f          ≤      k      −      2        (  x  ), where    λ  ∈                    F              q    ∗  ,    δ  ∈                    F              q    ∖  {      x    1    ,  ⋯  ,      x          n      −      1        },    ν  ∈                    F              q   and        f          ≤              k        −        2              (  x  )  ∈                    F              q    [  x  ] of a degree that does not exceed    k  −  2. This expands the result shown in our previous paper. In particular, we also show that the received word        u    =  (      u    1    ,  ⋯  ,      u          n        )  ∈                    F              q          n       is a deep hole of    C  (  D  ,  k  ) if and only if the Lagrange interpolation polynomial of the first    n  −  1 components of        u   is    λ  (  x  −  δ      )          q      −      2        +      u          n            x          k      −      1        +      u          ≤              k        −        2              (  x  )  , where    λ  ∈                    F              q    ∗  ,    δ  ∈                    F              q    ∖  {      x    1    ,  ⋯  ,      x          n      −      1        }, and        u          ≤              k        −        2              (  x  ) is a polynomial over                      F              q   whose degree does not exceed    k  −  2, if              q      −      1        2    ≤  k  <  n  ≤  q  −  2.}
}