Optimal bounds on codes for location in circulant graphs

Abstract

Identifying and locating-dominating codes have been studied widely in circulant graphs of type Cn(1, 2, 3, . . . , r) over the recent years. In 2013, Ghebleh and Niepel studied locatingdominating and identifying codes in the circulant graphs Cn(1, d) for d = 3 and proposed as an open question the case of d > 3. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs Cn(1, d), Cn(1, d − 1, d) and Cn(1, d − 1, d, d + 1). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in Cn(1, 3) and Cn(1, 4).

Improved Lower Bound for Locating-Dominating Codes in Binary Hamming Spaces

“…then according to Lemma 4 (in the cases n ≥ 13) together with Inequality (7) (in the cases n ∈ [11,12]) c is a special codeword, and further by Lemma 8 c is adjacent to exactly one special father, say u, which is actually sparse due to Lemma 9. We will first confirm that after applying Rule 2 and Rule 3 codeword c has the desired share.…”

“…Indeed, the inequality can be shown as follows. The cases n ∈ [11,13] are verified by substituting the corresponding value of n to the inequalities. When n ≥ 14, we observe that 1/(n − 2) + (1/6 + 1/(n − 2) − 1/(n − 5))/(n − 3) < 1/(n − 2) + 1/(6(n − 3)) < 1/10 and 1/6 + 1/(n 2 − 5n) − 2/(3n) > 1/6 − 2/(3n) > 1/10.…”

“…Now, we may apply Rule 2 to shift share to c. We shift h(1/6 + 1/(n − 2) − 1/(n − 5))/(n − 3) share to c. To ease the approximation, we notice that h(1/6 + 1/(n − 2) − 1/(n − 5))/(n − 3) < 1/6 since h ≤ n − 3 and we use this when n ≥ 13. In the following, the cases n ∈ [11,12] are verified by substituting the corresponding value of n to the inequalities Moreover, we have n/2 + 5/6 + 1/(n − 1) ≤ n/2 + 1 + 1/(n 2 − 5n) − 2/(3n) when n ≥ 13. Indeed, when n ≥ 13, we have 1/(n − 1) + 2/(3n) ≤ 1/6.…”

“…Let us assume that we have applied Rule 1 on the locating-dominating code C ⊆ F n . Observe that when n ∈ [11,12], we have n/2 + 1/2 + (n − 1)/(3(n − 4)) > n/2 + 1 − 1/(n − 1) and when n ≥ 13, we have n…”

“…Now, since the sensor placement is done using the locating-dominating code, we can deduce the location of the object just by considering which sensors are sending the alarm. The topic of locating-dominating codes has attracted a lot of attention recently, see [3,5,7,10,11]. For more papers in the field consult, the bibliography [13].…”