#### Abstract

Singleton-type upper bounds on the minimum Lee distance of general (not necessarily linear) Lee codes over are discussed. Two bounds known for linear codes are shown to also hold in the general case, and several new bounds are established. Codes meeting these bounds are investigated and in some cases characterised.

#### 1. Introduction

The Lee metric was introduced by Lee [1] in 1958 as an alternative to the Hamming metric for certain noisy channels. It found application and in particular was later developed for certain noisy channels (primarily those using phase-shift keying modulation [2]). The past decade has witnessed a burst of new and varied applications for codes defined in the Lee metric (Lee codes) including constrained and partial-response channels [3], interleaving schemes [4], orthogonal frequency-division multiplexing [5], multidimensional burst-error correction [6], and error correction for flash memories [7]. These recent applications give increased interest in questions surrounding optimal Lee codes.

Similar to the case of the Hamming metric, it is desirable to investigate upper bounds on the minimum Lee distance of a code given the code size, code length, and alphabet size. Codes meeting these bounds are of special interest as they are optimal in the sense that their minimum distance is largest. Under the Hamming metric, such codes are referred to as maximum distance separable (MDS) codes. Under the Lee metric, such codes may be referred to as *Maximum Lee Distance Separable (MLDS)* codes. Here, we will present several upper bounds similar to the Singleton bound and investigate the existence question of MLDS codes. In certain cases, we are able to completely characterize MLDS codes.

#### 2. Preliminaries

An block code is a collection of -tuples (codewords) over an alphabet of size such that the minimum (Hamming) distance between any two codewords is (hence, no two codewords have as many as common coordinates). Here, is the dimension of , which need not be an integer. Where context demands, we may also denote the Hamming distance by . The Singleton bound states that and holds for all block codes. Codes meeting this bound with equality are called maximum distance separable (MDS) codes. Research on both linear and nonlinear MDS codes has been extensive (e.g., see [8–10] and references therein).

##### 2.1. Lee Codes

Let be the set of representatives of the integer equivalence classes modulo . The *Lee weight* of any element is given by . Given an element , the *Lee weight* of , denoted , is given by
For , the *Lee distance * between and is defined to be the Lee weight of their difference,

A Lee code will be specified by , where , , and are as aforementioned, and is the *minimum Lee distance* of ; that is, .

Observe that in the case that or (binary or ternary codes), the Hamming and Lee metrics are identical. Thus a code with or is MLDS if and only if is MDS. As much literature discusses MDS codes, we shall focus attention on the cases .

A code over the alphabet is considered *linear* if forms a submodule of . Unless otherwise stated, we do not assume linearity.

##### 2.2. Code Equivalence

We shall say a permutation of is a *Lee permutation* if it preserves Lee distance. That is, , for all . For example, any translation is a Lee permutation.

Given an Lee code , we may define the following operations on codewords. A *positional permutation* is a permutation (on letters) of the coordinate positions, applied to each word of . A *Lee symbol permutation* is a Lee permutation applied to a fixed coordinate position throughout . Two codes are *equivalent* if one may be obtained from the other by applying a sequence of Lee symbol or positional permutations.

*Example 1. *The code
is a Lee code. The equivalent codes may be produced by applying the Lee permutation to one or more of the coordinate entries of . For example, the code
is equivalent to .

It follows that any given Lee Code is equivalent to a Lee Code containing the zero codeword. This observation shall prove quite useful in the sequel.

#### 3. Singleton-Type Bounds

In 2000 Shiromoto, [11] proved the following upper bound for linear codes over .

Proposition 2 (see [11]). *If is a linear Lee code over , then
*

We shall show that bound (6) holds for general (not necessarily linear) Lee codes. Indeed, observe that for a given code, any two codewords differ in at most coordinate positions, from which it follows that . Combining this observation with the Singleton bound (1) gives the following.

Proposition 3. *For any Lee code , the following holds:
*

*Remark 4. *Observe that
Consequently, Proposition 3 subsumes the bound of Shiromoto (6).

*Remark 5. *Suppose that is an Lee code with meeting bound (7), so that . Assume that contains the zero codeword. It follows that contains precisely two codewords. Moreover, the nonzero codeword has (Lee) weight . Hence, if is even, then the nonzero codeword has all entries . If is odd, then the nonzero codeword has each entry being either or .

We shall make improvements to bound (7) under various settings. Observe that codes for which necessarily contain repeated codewords; whence, . Also, codes with necessarily comprise the entire space , and consequently . Hence, in the sequel, we shall focus on codes with .

##### 3.1. Bounds on Codes over Even-Sized Alphabets

By restricting to codes over even-sized alphabets, we obtain the following bound.

Theorem 6. *For any Lee metric code with alphabet size even, of length and minimum Lee distance , the following holds:
*

*Proof. * If is even, we can create a distance preserving map (Gray code) (similar to that used in [12, 13]) from the metric space to the metric space defined by
For , we extend the map coordinatewise; thus, , and let . It follows that is a binary code of length with , and . The result follows from the Singleton bound (1).

###### 3.1.1. Special Cases: Characterizing and Improving

If restrictions are made on the size of , we are able to characterize cases of equality in bound (9). The following well known property of binary MDS codes shall be of use. See, for example, [14].

Lemma 7. *If is a binary MDS code of integral dimension , then either *(1)*, and is the repetition code , *(2)*, , and is the binary parity check code consisting of all binary -tuples of even (Hamming) weight, or *(3)*, , and is the entire space . *

Lemma 8. *Let be an Lee code with , even, and . If , then either *(1)*, and is the repetition code , or *(2)*, and . *

*Proof. * Assume that . Let be as defined in the proof of Theorem 6. Then, is a binary MDS code, , and . Note that since and , we have . According to Lemma 7, we have three cases to consider.*Case * 1 . In this case, is the binary repetition code; hence, is the code
*Case* 2 . In this case, is the binary parity check code of length . As such, the vector is in . From the definition of , it follows that ; so, .

Note that since , we have .*Case* 3 . In this case, . Again, the vector is in , giving . But then, , contradicting .

*Remark 9. *An example of a code satisfying item in Lemma 8 is the Lee code:

With Lemma 8, we see that the bound in Theorem 6 may be improved in certain cases.

Theorem 10. *Let be an Lee code with even, , and . Then,
*

##### 3.2. Bounds on General Alphabets

In this section, we shall make improvements to the bound in Proposition 3. First, some intermediary lemmata are introduced.

Lemma 11. *If is an Lee code with , then the following hold:*(1)* is an integer;*(2)*Fix coordinate positions , and (not necessarily distinct) elements . Then, there exists precisely one codeword having , . *

*Proof. *For the first part, observe that if two codewords agree in coordinates, then their distance is at most , violating the minimum Lee distance condition. Consequently, , giving .

For the second part, fix coordinate positions . The result then follows by observing that no two codewords agree in all of these positions, and there are codewords in total.

Lemma 12. *If is a Lee code, , then . *

*Proof. *First, note that from Example 1, a Lee code does in fact exist.

Now, let , and assume without loss of generality that the zero codeword is in . Consider (Lemma 11) the three codewords , , and , having first coordinates , , and , respectively. In order to maintain minimum Lee distance from the zero codeword, the -st coordinate of each must be from . Thus, at least two of agree in the final coordinate, violating the minimum distance property.

Lemma 13. *If is an code with , and integral dimension , then . *

*Proof. * Suppose that is an code, , , with . From Lemma 11, there exist codewords , and having first coordinates and th coordinate , , and , respectively. Consequently, we have .

If is even, then gives , for . Similarly, , for . But then, , contradicting the condition .

If is odd, then a similar argument shows that for each of the final coordinate, . Consequently, . Therefore, we have from which it follows that , , and . From Lemma 12, it then follows that , a contradiction.

From Lemmas 11 and 13, we have the following result.

Theorem 14. *If is an Lee code , then
*

##### 3.3. Bound Based on Plotkin’s Average Distance Bound

Based on the fact that the minimum distance between pairs of codewords cannot exceed the average distance between all pairs of distinct codewords, Wyner and Graham [15] obtained the following bound.

Proposition 15. *If is an Lee code, then
**
where is the average Lee weight of , given by
*

Using the previous result, we may establish a further Singleton-type bound for Lee codes.

Theorem 16. *Let be an code. Let be the maximal number of codewords mutually agreeing in common coordinates. Then,
**
where
*

*Proof. * Let be an code, and let be as defined earlier. Consider a collection of codewords mutually agreeing in coordinates, assumed to be the first coordinates. Let be the corresponding punctured code obtained by deleting the first coordinates of each word in .

It follows that is an code with . Applying Proposition 15 to , we obtain

Finally, applying (16), we get the result.

Suppose that is an Lee code, and is as defined ealier. Observe that since , it follows that . So, if is even, the bound in Theorem 16 always meets or exceeds bound (7). Moreover, if , simple counting shows that . This gives the following corollary, much improving bound (7).

Corollary 17. *If is an Lee code with , then
**
where
*

*Remark 18. *Chiang and Wolf [16] established the bound in Corollary 17 for linear codes.

Codes meeting the bounds in Theorem 16 do exist [16]. Two examples follow: (1)for as prime, the linear one-dimensional (equidistant) code generated by ; (2)the linear Lee code with generator matrix

#### 4. Conclusion

Several upper bounds on the minimum Lee distance of a block code similar to the Singleton bound were established. Codes actually meeting some of these bounds were presented. Two bounds known for linear codes were shown (see Proposition 2 and Corollary 17) to hold for nonlinear codes. Several new bounds were also established.

#### Acknowledgments

The first author thanks the NSERC of Canada for support through the Discovery Grants Program. The second author thanks the NSERC of Canada for support through the Undergraduate Student Research Awards Program.