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Construction of CCC and ZCCS Through Additive Characters Over Galois Field

Gobinda Ghosh, Sachin Pathak

TL;DR

This paper tackles the limitation of code-length versatility in complete complementary codes (CCC) and Z-complementary code sets (ZCCS) for MC-CDMA by introducing an additive-character framework over finite Galois fields GF($p^{r}$). It first constructs CCCs of length $p^{r}$ and then extends to ZCCS of length $n p^{r}$ with arbitrary positive integers $n$ and $r$, showing that existing ZCCS lengths are encompassed as special cases. The core method uses additive characters and trace functions to define sequences over GF($q$) that achieve optimal zero-correlation zones, with explicit constructions and examples (including a $q=9$ CCC and an $(18,9,9,18)$ ZCCS). The proposed framework broadens the available lengths and user capacities for ZCCS, offering a versatile approach to interference reduction in MC-CDMA systems and aligning with prior literature while surpassing previous length constraints.

Abstract

The rapid progression in wireless communication technologies, especially in multicarrier code-division multiple access (MC-CDMA), there is a need of advanced code construction methods. Traditional approaches, mainly based on generalized Boolean functions, have limitations in code length versatility. This paper introduces a novel approach to constructing complete complementary codes (CCC) and Z-complementary code sets (ZCCS), for reducing interference in MC-CDMA systems. The proposed construction, distinct from Boolean function-based approaches, employs additive characters over Galois fields GF($p^{r}$), where $p$ is prime and $r$ is a positive integer. First, we develop CCCs with lengths of $p^{r}$, which are then extended to construct ZCCS with both unreported lengths and sizes of $np^{r}$, where $n$ are arbitrary positive integers. The versatility of this method is further highlighted as it includes the lengths of ZCCS reported in prior studies as special cases, underscoring the method's comprehensive nature and superiority.

Construction of CCC and ZCCS Through Additive Characters Over Galois Field

TL;DR

This paper tackles the limitation of code-length versatility in complete complementary codes (CCC) and Z-complementary code sets (ZCCS) for MC-CDMA by introducing an additive-character framework over finite Galois fields GF(). It first constructs CCCs of length and then extends to ZCCS of length with arbitrary positive integers and , showing that existing ZCCS lengths are encompassed as special cases. The core method uses additive characters and trace functions to define sequences over GF() that achieve optimal zero-correlation zones, with explicit constructions and examples (including a CCC and an ZCCS). The proposed framework broadens the available lengths and user capacities for ZCCS, offering a versatile approach to interference reduction in MC-CDMA systems and aligning with prior literature while surpassing previous length constraints.

Abstract

The rapid progression in wireless communication technologies, especially in multicarrier code-division multiple access (MC-CDMA), there is a need of advanced code construction methods. Traditional approaches, mainly based on generalized Boolean functions, have limitations in code length versatility. This paper introduces a novel approach to constructing complete complementary codes (CCC) and Z-complementary code sets (ZCCS), for reducing interference in MC-CDMA systems. The proposed construction, distinct from Boolean function-based approaches, employs additive characters over Galois fields GF(), where is prime and is a positive integer. First, we develop CCCs with lengths of , which are then extended to construct ZCCS with both unreported lengths and sizes of , where are arbitrary positive integers. The versatility of this method is further highlighted as it includes the lengths of ZCCS reported in prior studies as special cases, underscoring the method's comprehensive nature and superiority.
Paper Structure (6 sections, 3 theorems, 33 equations, 4 figures, 1 table)

This paper contains 6 sections, 3 theorems, 33 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Characters on Finite Fields
  4. Constructions of CCC
  5. Constructions of ZCCS
  6. Conclusion

Key Result

Lemma 1

liu2011correlation For a $(s,m,l,z)$-ZCCS configuration, it is established that $s \leq m \left\lfloor \frac{l}{z} \right\rfloor$, where $s$, $m$, $l$, and $z$ represent the number of users, the number of sub-carriers, the code length, and the ZCZ width, respectively. The ZCCS is considered optimal where $\left\lfloor . \right\rfloor$ denotes floor function.

Figures (4)

  • Figure 1: Auto-correlation result of any set of array from $\mathcal{S}$
  • Figure 2: Cross-correlation result of any set of array from $\mathcal{S}$
  • Figure 3: Auto-correlation result of any set of array from $T$
  • Figure 4: Cross-correlation result of any set of array from $T$

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Example 2
  • ...and 1 more