od: report current offset for empty strings with --strings=0

src/uu/od/src/od.rs

@@ -632,7 +632,7 @@ fn extract_strings_from_input(
     };
 
     let mut current_string = Vec::new();
-    let mut string_start_offset = 0u64;
+    let mut string_start_offset = skip_bytes;
     let mut current_offset = skip_bytes;
     let mut bytes_read = 0u64;
     let mut buf = [0u8; 1];
@@ -665,7 +665,12 @@ fn extract_strings_from_input(
                     // Either null terminator or non-printable character
                     if byte == 0 && current_string.len() >= min_length {
                         // Null terminator found with valid string
-                        print_string(string_start_offset, &current_string)?;
+                        let offset = if current_string.is_empty() {
+                            current_offset
+                        } else {
+                            string_start_offset
+                        };
+                        print_string(offset, &current_string)?;
                     }
                     current_string.clear();
                 }

tests/by-util/test_od.rs

@@ -1390,6 +1390,25 @@ fn test_od_strings_option() {
         .succeeds()
         .stdout_is("0000000 hello\n0000006 world\n");
 
+    // Test -S0 with empty strings and consecutive NUL bytes (issue #16134)
+    new_ucmd!()
+        .args(&["--strings=0", "-Ad"])
+        .pipe_in(b"abc\x00\x00def\x00")
+        .succeeds()
+        .stdout_is("0000000 abc\n0000004 \n0000005 def\n");
+
+    new_ucmd!()
+        .args(&["-S0", "-Ad"])
+        .pipe_in(b"\x00\x00")
+        .succeeds()
+        .stdout_is("0000000 \n0000001 \n");
+
+    new_ucmd!()
+        .args(&["--strings=0", "-Ad", "-j4"])
+        .pipe_in(b"abc\x00\x00def\x00")
+        .succeeds()
+        .stdout_is("0000004 \n0000005 def\n");
+
     // Test -S0 with single character strings
     new_ucmd!()
         .arg("-S0")