# Bounds for Coding Theory over Rings

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{s}for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight.

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- 1.
- $w\left(0\right)=0$ and $w\left(x\right)>0$ for all $x\ne 0$,
- 2.
- $w\left(x\right)=w(-x)$ for all $x\in R$,
- 3.
- $w(x+y)\le w\left(x\right)+w\left(y\right)$ for all $x,y\in R$,

- 1.
- $d(x,y)\ge 0$ for all $x,y\in R$ and $d(x,y)=0$ if and only if $x=y$.
- 2.
- $d(x,y)=d(y,x)$ for all $x,y\in R$,
- 3.
- $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in R$.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

- (i)
- For all $x,y$ with $Rx=Ry$, we have that $w\left(x\right)=w\left(y\right)$.
- (ii)
- For every non-zero ideal $I\le {}_{R}R$, it holds that$$\frac{1}{\left|I\right|}\sum _{x\in I}w\left(x\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\gamma .$$

**Theorem**

**1**

**Theorem**

**2**

## 3. Overweight

**Definition**

**7.**

#### Relations to Other Weights

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Lemma**

**1.**

## 4. Bounds for the Overweight

**Definition**

**8.**

#### 4.1. A Singleton Bound

**Remark**

**1.**

**Proposition**

**2.**

**Example**

**2.**

**Proposition**

**3.**

**Example**

**3.**

#### 4.2. A Sphere-Packing Bound

**Definition**

**9.**

**Lemma**

**2.**

**Corollary**

**1**

**Corollary**

**2.**

**Proof.**

#### 4.3. A Gilbert–Varshamov Bound

**Proposition**

**4**

**Proof.**

**Example**

**4.**

#### 4.4. A Plotkin Bound

**Lemma**

**3.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Theorem**

**3**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Remark**

**2.**

**Example**

**5.**

## 5. A Johnson Bound for the Homogeneous Weight

**Definition**

**10.**

**Definition**

**11.**

**Proposition**

**7**

**Theorem**

**4.**

- (i)
- We have that $\gamma \phantom{\rule{0.166667em}{0ex}}n(d-\gamma \phantom{\rule{0.166667em}{0ex}}n)\ge 1$.
- (ii)
- It holds that $\rho \le \gamma -\sqrt{(\gamma -\frac{d}{n})\gamma +\frac{1}{{n}^{2}}}$.

**Proof.**

**Remark**

**3.**

**Example**

**6.**

## 6. Open Problems

**Problem**

**1.**

**Problem**

**2.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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${\mathit{w}}_{\mathit{H}}$ | $\mathit{wt}$ | ${\mathit{w}}_{\mathit{L}}$ | ${\mathit{w}}_{\mathit{K}}$ | W | |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 1/2 | 1 | 1 | 1 |

2 | 1 | 3/2 | 2 | 2 | 2 |

3 | 1 | 2 | 3 | 1 | 2 |

4 | 1 | 3/2 | 2 | 2 | 2 |

5 | 1 | 1/2 | 1 | 1 | 1 |

${\mathit{w}}_{\mathit{H}}$ | $\mathit{wt}$ | ${\mathit{w}}_{\mathit{B}}$ | W | |
---|---|---|---|---|

(0,0) | 0 | 0 | 0 | 0 |

(0,1) | 1 | 2 | 2 | 2 |

(1,0) | 1 | 2 | 2 | 2 |

(1,1) | 2 | 0 | 1 | 1 |

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**MDPI and ACS Style**

Gassner, N.; Greferath, M.; Rosenthal, J.; Weger, V.
Bounds for Coding Theory over Rings. *Entropy* **2022**, *24*, 1473.
https://doi.org/10.3390/e24101473

**AMA Style**

Gassner N, Greferath M, Rosenthal J, Weger V.
Bounds for Coding Theory over Rings. *Entropy*. 2022; 24(10):1473.
https://doi.org/10.3390/e24101473

**Chicago/Turabian Style**

Gassner, Niklas, Marcus Greferath, Joachim Rosenthal, and Violetta Weger.
2022. "Bounds for Coding Theory over Rings" *Entropy* 24, no. 10: 1473.
https://doi.org/10.3390/e24101473