On Maximum Lee Distance Codes

The Lee metric was introduced by Lee [1] in 1958 as an alternative to theHammingmetric 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 partialresponse 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.


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 partialresponse 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.

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 − + 1 common coordinates). Here, = log | | 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][9][10] and references therein).

Lee Codes.
Let Z = {0, 1, . . . , − 1} be the set of representatives of the integer equivalence classes modulo . The Lee weight of any element ∈ Z is given by ( ) = min{ , − }. Given an element = ( 1 , 2 , . . . , ) ∈ Z , the Lee weight of , denoted ( ), is given by For , ∈ Z , the Lee distance ( , ) between and is defined to be the Lee weight of their difference,

Journal of Discrete Mathematics
A Lee code will be specified by ( , , ) , where , , and are as aforementioned, and is the minimum Lee distance of ; that is, = min{ ( , ) | , ∈ , ̸ = }. Observe that in the case that = 2 or 3 (binary or ternary codes), the Hamming and Lee metrics are identical. Thus a code with = 2 or = 3 is MLDS if and only if is MDS. As much literature discusses MDS codes, we shall focus attention on the cases > 3.
A code over the alphabet Z is considered linear if forms a submodule of Z . Unless otherwise stated, we do not assume linearity.

Code Equivalence.
We shall say a permutation of Z is a Lee permutation if it preserves Lee distance. That is, ( , ) = ( ( ), ( )), for all , ∈ Z . For example, any translation ( ) = + (mod ) 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.
is a (2, 1, 3) 5 Lee code. The equivalent codes may be produced by applying the Lee permutation (0)(14)(23) 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.

Singleton-Type Bounds
In 2000 Shiromoto, [11] proved the following upper bound for linear codes over Z .
Proposition 2 (see [11]). If is a linear ( , , ) Lee code over Z , 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 ≤ ⌊ /2⌋ . Combining this observation with the Singleton bound (1) gives the following.
We shall make improvements to bound (7) under various settings. Observe that codes for which < necessarily contain repeated codewords; whence, = 0. Also, codes with = necessarily comprise the entire space Z , and consequently = 1. Hence, in the sequel, we shall focus on codes with > .

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 (Z , ) to the metric space (Z /2 2 , ) defined by For ∈ , we extend the map coordinatewise; thus, ( ) = ( ( 1 ) ⋅ ⋅ ⋅ ( )), and let = ( ) = { ( ) | ∈ }. It follows that is a binary code of length ( /2) with | | = | |, and ( ) = ( ). The result follows from the Singleton bound (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]. Proof. Assume that = ( /2) − log 2 | | + 1. Let be as defined in the proof of Theorem 6. Then, = ( ) is a binary ( , , ) MDS code, = ( /2) , and = log 2 | |. Note that since ≥ 2 and ≥ 4, we have ≥ 4. According to Lemma 7, we have three cases to consider.
Case 1 ( = 1). In this case, is the binary repetition code; hence, is the code Case 2 ( > 1, = +1). In this case, is the binary parity check code of length . As such, the vector (101000 ⋅ ⋅ ⋅ 0) is in . From the definition of , it follows that /2 ≤ 2; so, = 4. Note that since = 4, we have = 2 = + 1 = 2 + 1. With Lemma 8, we see that the bound in Theorem 6 may be improved in certain cases.

Bounds on General Alphabets.
In this section, we shall make improvements to the bound in Proposition 3. First, some intermediary lemmata are introduced.
For the second part, fix coordinate positions { 1 , . . . , }. The result then follows by observing that no two codewords agree in all of these positions, and there are codewords in total.
From Lemmas 11 and 13, we have the following result.

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 Z , given by Using the previous result, we may establish a further Singleton-type bound for Lee codes.
Proof. Let be an ( , , ) code, and let be as defined earlier. Consider a collection 1 of codewords mutually agreeing in ⌈ ⌉ − 1 coordinates, assumed to be the first ⌈ ⌉ − 1 coordinates. Let be the corresponding punctured code obtained by deleting the first ⌈ ⌉ − 1 coordinates of each word in 1 .
Suppose that is an ( , , ) Lee code, and is as defined ealier. Observe that since | | > ⌈ ⌉−1 , it follows that ≥ 2. So, if is even, the bound in Theorem 16 always meets or exceeds bound (7). Moreover, if ∈ Z, simple counting shows that ≥ . This gives the following corollary, much improving bound (7).
Remark 18. Chiang and Wolf [16] established the bound in Corollary 17 for linear codes.

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.