Correction to: Good integers and some applications in coding theory

Correction to: Cryptogr. Commun. (2018) 10:685–704

https://doi.org/10.1007/s12095-017-0255-4

Proposition 2.1 and Proposition 2.3 in the original publication are incorrectly worded and they should be as follows.

FormalPara Proposition 2.1

Let a and b be coprime odd integers and letβ ≥ 1 be an integer. Then the following statements are equivalents.

• i) 2^β ∈ G_(a,b).

• ii) 2^β|(a + b).

• iii) ab⁻¹ ≡ −1 mod 2^β.

FormalPara Proposition 2.3

Let a,b and d > 1 be pairwise coprime odd positive integers and letβ ≥ 2 be an integer. Then 2^βd ∈ G_(a,b) if and only if 2^β|(a + b) and d ∈ G_(a,b) is such that $2|| \text {ord}_{d}(\frac {a}{b})$. In this case, $\text {ord}_{2^{\beta }}(\frac {a}{b})= 2$ and $2|| \text {ord}_{2^{\beta } d}(\frac {a}{b})$.

As a consequence of the above corrections, the bullets (c) and (d) of Theorem 2.1 and Theorem 3.1 in the original paper should be rewritten as follows.

• (c) β ≥ 2,d = 1 and 2^β|(a + b).

• (d) β ≥ 2,d ≥ 3, 2^β|(a + b) and d ∈ G_(a,b) is such that $2|| \text {ord}_{d}(\frac {a}{b})$.

The above rewordings do not affect any other result given in the paper.

The readers may refer to http://www.math.sc.su.ac.th/web3/files/somphong/J-Correction-GoodIntegers.pdf for a full discussion.