On Maximum Lee Distance Codes

Abstract

Singleton-type upper bounds on the minimum Lee distance of general (not necessarily linear) Lee codes over ℤ_𝑞 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 (𝑛,𝑘,𝑑)_𝑞 block code 𝐶 is a collection of 𝑞^𝑘𝑛-tuples (codewords) over an alphabet of size 𝑞 such that the minimum (Hamming) distance between any two codewords is 𝑑 (hence, no two codewords have as many as 𝑛−𝑑+1 common coordinates). Here, 𝑘=log_𝑞|𝐶| is the dimension of 𝐶, which need not be an integer. Where context demands, we may also denote the Hamming distance by 𝑑_𝐻. The Singleton bound states that 𝑑≤𝑛−⌈𝑘⌉+1 (1) 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 ℤ_𝑞={0,1,…,𝑞−1} be the set of representatives of the integer equivalence classes modulo 𝑞. The Lee weight of any element 𝑎∈ℤ_𝑞 is given by 𝑤_𝐿(𝑎)=min{𝑎,𝑞−𝑎}. Given an element 𝑐=(𝑐₁,𝑐₂,…,𝑐_𝑛)∈ℤ^𝑛_𝑞, the Lee weight of 𝑐, denoted 𝑤_𝐿(𝑐), is given by 𝑤_𝐿(𝑐)=^𝑛_𝑖=1min𝑎,𝑞−𝑎. (2) For 𝑐,𝑒∈ℤ^𝑛_𝑞, the Lee distance 𝑑_𝐿(𝑐,𝑒) between 𝑐 and 𝑒 is defined to be the Lee weight of their difference, 𝑑_𝐿(𝑐,𝑒)=𝑤_𝐿()=𝑐−𝑒_𝑛𝑖=1min𝑐_𝑖−𝑒_𝑖mod𝑞,𝑒_𝑖−𝑐_𝑖mod𝑞. (3)

A Lee code 𝐶 will be specified by (𝑛,𝑘,𝑑_𝐿)_𝑞, where 𝑛, 𝑘, and 𝑞 are as aforementioned, and 𝑑_𝐿 is the minimum Lee distance of 𝐶; that is, 𝑑_𝐿=min{𝑑_𝐿(𝑐,𝑒)∣𝑐,𝑒∈𝐶,𝑐≠𝑒}.

Observe that in the case that 𝑞=2 or 3 (binary or ternary codes), the Hamming and Lee metrics are identical. Thus a code 𝐶 with 𝑞=2 or 𝑞=3 is MLDS if and only if 𝐶 is MDS. As much literature discusses MDS codes, we shall focus attention on the cases 𝑞>3.

A code 𝐶 over the alphabet ℤ_𝑞 is considered linear if 𝐶 forms a submodule of ℤ^𝑛_𝑞. Unless otherwise stated, we do not assume linearity.

2.2. Code Equivalence

We shall say a permutation 𝜎 of ℤ_𝑞 is a Lee permutation if it preserves Lee distance. That is, 𝑑_𝐿(𝑎,𝑏)=𝑑_𝐿(𝜎(𝑎),𝜎(𝑏)), for all 𝑎,𝑏∈ℤ_𝑞. For example, any translation 𝜎(𝑥)=𝑥+𝑡(mod𝑞) is a Lee permutation.

Given an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code 𝐶, we may define the following operations on codewords. A positional permutation is a permutation (on 𝑛 letters) of the coordinate positions, applied to each word of 𝐶. A Lee symbol permutation is a Lee permutation applied to a fixed coordinate position throughout 𝐶. 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 𝐶=(0,0)(1,2)(2,4)(3,1)(4,3) (4) 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 𝐶. For example, the code 𝐶^′=(0,0)(1,3)(2,1)(3,4)(4,2) (5) is equivalent to 𝐶.

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 ℤ_𝑞.

Proposition 2 (see [11]). If 𝐶 is a linear (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code over ℤ_𝑞, then 𝑑_𝐿−1𝑞/2≤𝑛−𝑘. (6)

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 𝑑_𝐻 coordinate positions, from which it follows that 𝑑_𝐿≤⌊𝑞/2⌋𝑑_𝐻. Combining this observation with the Singleton bound (1) gives the following.

Proposition 3. For any (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code 𝐶, the following holds: 𝑑_𝐿≤𝑞2(𝑛−⌈𝑘⌉+1). (7)

Remark 4. Observe that 𝑑_𝐿≤𝑞2⌈𝑘⌉)⇒𝑑(𝑛−+1^𝐿−1≤⌈𝑘⌉+𝑞/2𝑛−𝑞/2−1⌈𝑘⌉.𝑞/2=𝑛− (8) Consequently, Proposition 3 subsumes the bound of Shiromoto (6).

Remark 5. Suppose that 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code with ⌈𝑘⌉=1 meeting bound (7), so that 𝑑_𝐿=⌊𝑞/2⌋𝑛. Assume that 𝐶 contains the zero codeword. It follows that 𝐶 contains precisely two codewords. Moreover, the nonzero codeword has (Lee) weight 𝑑_𝐿. Hence, if 𝑞 is even, then the nonzero codeword has all entries 𝑞/2. If 𝑞 is odd, then the nonzero codeword has each entry being either (𝑞+1)/2 or (𝑞−1)/2.

We shall make improvements to bound (7) under various settings. Observe that codes for which 𝑛<𝑘 necessarily contain repeated codewords; whence, 𝑑_𝐿=0. Also, codes with 𝑛=𝑘 necessarily comprise the entire space ℤ^𝑛_𝑞, and consequently 𝑑_𝐿=1. Hence, in the sequel, we shall focus on codes with 𝑛>𝑘.

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 𝑞 even, of length 𝑛 and minimum Lee distance 𝑑_𝐿, the following holds: 𝑑_𝐿≤𝑞2𝑛−log₂|𝐶|+1. (9)

Proof. If 𝑞 is even, we can create a distance preserving map (Gray code) 𝜙 (similar to that used in [12, 13]) from the metric space (ℤ_𝑞,𝑑_𝐿) to the metric space (ℤ₂^𝑞/2,𝑑_𝐻) defined by 𝜙(𝑎)=^{𝑞/2−𝑎}000⋯^𝑎𝑞111if0≤𝑎≤2−1,_𝑞−𝑎111⋯^{𝑎−𝑞/2}𝑞000if2≤𝑎≤𝑞−1. (10) For 𝑐∈𝐶, we extend the map coordinatewise; thus, 𝜙(𝑐)=(𝜙(𝑐₁)⋯𝜙(𝑐_𝑛)), and let 𝐶^′=𝜙(𝐶)={𝜙(𝑐)∣𝑐∈𝐶}. It follows that 𝐶^′ is a binary code of length 𝑛(𝑞/2) with |𝐶|=|𝐶^′|, and 𝑑_𝐿(𝐶)=𝑑_𝐻(𝐶^′). The result follows from the Singleton bound (1).

3.1.1. Special Cases: Characterizing and Improving

If restrictions are made on the size of 𝐶, 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 𝐶 is a binary (𝑛,𝑘,𝑑)₂ MDS code of integral dimension 𝑘≥1, then either (1)𝑘=1, and 𝐶 is the repetition code {(0,0,…,0),(1,1,…,1)}, (2)𝑘≥2, 𝑛=𝑘+1, and 𝐶 is the binary parity check code consisting of all binary 𝑛-tuples of even (Hamming) weight, or (3)𝑘≥2, 𝑛=𝑘, and 𝐶 is the entire space ℤ^𝑛₂.

Lemma 8. Let 𝐶 be an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code with 𝑛>𝑘≥1, 𝑞>2 even, and log₂|𝐶|∈ℤ. If 𝑑_𝐿=(𝑞/2)𝑛−log₂|𝐶|+1, then either (1)𝑘=1, and 𝐶 is the repetition code {(00⋯0)((𝑞/2)(𝑞/2)⋯𝑞/2)}, or (2)𝑘=𝑛−(1/2), and 𝑞=4.

Proof. Assume that 𝑑_𝐿=(𝑞/2)𝑛−log₂|𝐶|+1. Let 𝜙 be as defined in the proof of Theorem 6. Then, 𝐶^′=𝜙(𝐶) is a binary (𝑛^′,𝑘^′,𝑑^′) MDS code, 𝑛^′=(𝑞/2)𝑛, and 𝑘^′=log₂|𝐶|. Note that since 𝑛≥2 and 𝑞≥4, we have 𝑛^′≥4. According to Lemma 7, we have three cases to consider.
Case  1 (𝑘^′=1). In this case, 𝐶^′ is the binary repetition code; hence, 𝐶 is the code ()𝑞00⋯02𝑞2⋯𝑞2. (11)
Case  2 (𝑘^′>1,𝑛^′=𝑘^′+1). In this case, 𝐶^′ is the binary parity check code of length 𝑛^′. As such, the vector (101000⋯0) is in 𝐶^′. From the definition of 𝜙, it follows that 𝑞/2≤2; so, 𝑞=4.
Note that since 𝑞=4, we have 𝑛^′=2𝑛=𝑘^′+1=2𝑘+1.
Case  3 (𝑘^′>1,𝑛^′=𝑘^′). In this case, 𝐶^′=ℤ^𝑘′₂. Again, the vector (101000⋯0) is in 𝐶^′, giving 𝑞=4. But then, 𝑛^′=2𝑛=𝑘^′=2𝑘, contradicting 𝑛>𝑘.

Remark 9. An example of a code satisfying item (2) in Lemma 8 is the (2,3/2,2)₄ Lee code: 𝐶={(00),(02),(20),(22),(13),(31),(11),(33)}. (12)

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

Theorem 10. Let 𝐶 be an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code with 𝑞>4 even, 𝑛>𝑘≥1, and log₂|𝐶|∈ℤ. Then, 𝑑_𝐿≤𝑞2𝑛−log₂|𝐶|. (13)

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 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code with 𝑑_𝐿≥⌊𝑞/2⌋(𝑛−⌊𝑘⌋)+1, then the following hold:(1)𝑘 is an integer;(2)Fix 𝑘 coordinate positions 𝑖₁<𝑖₂<⋯<𝑖_𝑘, and 𝑘 (not necessarily distinct) elements 𝛼₁,…,𝛼_𝑘∈ℤ_𝑞. Then, there exists precisely one codeword 𝑐 having 𝑐_𝑖𝑗=𝛼_𝑗, 𝑗=1,2,…,𝑘.

Proof. For the first part, observe that if two codewords agree in ⌊𝑘⌋ coordinates, then their distance is at most ⌊𝑞/2⌋(𝑛−⌊𝑘⌋), violating the minimum Lee distance condition. Consequently, |𝐶|≤𝑞^⌊𝑘⌋, giving 𝑘=⌊𝑘⌋.
For the second part, fix 𝑘 coordinate positions {𝑖₁,…,𝑖_𝑘}. The result then follows by observing that no two codewords agree in all of these positions, and there are 𝑞^𝑘 codewords in total.

Lemma 12. If 𝐶 is a (𝑘+1,𝑘,3)₅ Lee code, 𝑘∈ℤ, then 𝑘=1.

Proof. First, note that from Example 1, a (2,1,3)₅ Lee code does in fact exist.
Now, let 𝑘≥2, and assume without loss of generality that the zero codeword is in 𝐶. Consider (Lemma 11) the three codewords 𝑤₁, 𝑤₂, and 𝑤₃, having first 𝑘 coordinates 100⋯0, 0100⋯0, and 400⋯0, respectively. In order to maintain minimum Lee distance 3 from the zero codeword, the (𝑘+1)-st coordinate of each 𝑤_𝑖 must be from {2,3}. Thus, at least two of {𝑤₁,𝑤₂,𝑤₃} agree in the final coordinate, violating the minimum distance property.

Lemma 13. If 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 code with 𝑞>3, and integral dimension 𝑘>1, then 𝑑_𝐿≤⌊𝑞/2⌋(𝑛−𝑘).

Proof. Suppose that 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 code, 𝑞>3, 1<𝑘∈ℤ, with 𝑑_𝐿≥⌊𝑞/2⌋(𝑛−𝑘)+1. From Lemma 11, there exist codewords 𝑎, 𝑏, and 𝑐 having first 𝑘−1 coordinates 0 and 𝑘th coordinate 0, 1, and 𝑞−1, respectively. Consequently, we have 𝑑_𝐿=⌊𝑞/2⌋(𝑛−𝑘)+1.
If 𝑞 is even, then 𝑑_𝐿(𝑎,𝑏)=(𝑞/2)(𝑛−𝑘)+1 gives 𝑏_𝑖=𝑎_𝑖+(𝑞/2)(mod𝑞), for 𝑖=𝑘+1,𝑘+2,…,𝑛. Similarly, 𝑐_𝑖=𝑎_𝑖+(𝑞/2)(mod𝑞), for 𝑖=𝑘+1,𝑘+2,…,𝑛. But then, 𝑑_𝐿(𝑏,𝑐)=2, contradicting the condition 𝑑_𝐿=(𝑞/2)(𝑛−𝑘)+1.
If 𝑞 is odd, then a similar argument shows that for each of the final (𝑛−𝑘) coordinate, 𝑑_𝐿(𝑏_𝑖,𝑐_𝑖)≤1. Consequently, 𝑑_𝐿(𝑏,𝑐)≤𝑛−𝑘+2. Therefore, we have 𝑛−𝑘+2≥⌊𝑞/2⌋(𝑛−𝑘)+1 from which it follows that 𝑞=5, 𝑛=𝑘+1, and 𝑑=3. From Lemma 12, it then follows that 𝑘=1, a contradiction.

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

Theorem 14. If 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code 𝑞>3, 𝑛>𝑘>1 then 𝑑_𝐿≤𝑞2(𝑛−⌊𝑘⌋). (14)

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 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code, then 𝑑_𝐿≤𝑛𝐷(|𝐶,1−1/|) (15) where 𝐷 is the average Lee weight of ℤ_𝑞, given by 𝑞𝐷=²−1𝑞4𝑞𝑖𝑓𝑞𝑖𝑠𝑜𝑑𝑑,4𝑖𝑓𝑞𝑖𝑠𝑒𝑣𝑒𝑛. (16)

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

Theorem 16. Let 𝐶 be an (𝑛,𝑘,𝑑_𝐿)_𝑞 code. Let 𝛼 be the maximal number of codewords mutually agreeing in ⌈𝑘⌉−1 common coordinates. Then, 𝑑_𝐿≤⌊(𝑛−⌈𝑘⌉+1)⋅𝑎⌋, (17) where 𝑞𝑎=²−1⋅𝛼4𝑞𝑞𝛼−1𝑖𝑓𝑞𝑜𝑑𝑑,4⋅𝛼𝛼−1𝑖𝑓𝑞𝑒𝑣𝑒𝑛. (18)

Proof. Let 𝐶 be an (𝑛,𝑘,𝑑_𝐿)_𝑞 code, and let 𝛼 be as defined earlier. Consider a collection 𝐶₁ of 𝛼 codewords mutually agreeing in ⌈𝑘⌉−1 coordinates, assumed to be the first ⌈𝑘⌉−1 coordinates. Let 𝐶^′ be the corresponding punctured code obtained by deleting the first ⌈𝑘⌉−1 coordinates of each word in 𝐶₁.
It follows that 𝐶^′ is an (𝑛−⌈𝑘⌉+1,1,𝑑^′_𝐿)_𝑞 code with 𝑑^′_𝐿≥𝑑_𝐿. Applying Proposition 15 to 𝐶^′, we obtain 𝑑_𝐿≤𝑑^′_𝐿≤(𝑛−⌈𝑘⌉+1)𝐷().1−1/𝛼 (19)
Finally, applying (16), we get the result.

Suppose that 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code, and 𝛼 is as defined ealier. Observe that since |𝐶|>𝑞^{⌈𝑘⌉−1}, it follows that 𝛼≥2. So, if 𝑞 is even, the bound in Theorem 16 always meets or exceeds bound (7). Moreover, if 𝑘∈ℤ, simple counting shows that 𝛼≥𝑞. This gives the following corollary, much improving bound (7).

Corollary 17. If 𝐶 is an (𝑛,𝑘,𝑑_𝐿)_𝑞 Lee code with 𝑘∈ℤ, then 𝑑_𝐿≤(𝑛−𝑘+1)⋅𝑎, (20) where 𝑞𝑎=4𝑞−²1𝑖𝑓𝑞𝑒𝑣𝑒𝑛,𝑞+14𝑖𝑓𝑞𝑜𝑑𝑑. (21)

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 𝑝 as prime, the linear one-dimensional (equidistant) (𝑝−1,1,((𝑝−1)/2)⋅3)_𝑝 code generated by (1,2,…,𝑝−1); (2)the linear (4,2,4)₅ Lee code with generator matrix 𝐺=34100341. (22)

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.